green's theorem clockwise

∫ ∫ D ∂ Q ∂ x − ∂ P ∂ y d A = ∫ C P d x + Q d y, provided the integration on the right is done counter-clockwise around C . Applying the two-dimensional divergence theorem with = (,), we get the right side of Green's theorem: ∮ C ( M , − L ) ⋅ n ^ d s = ∬ D ( ∇ ⋅ ( M , − L ) ) d A = ∬ D ( ∂ M ∂ x − ∂ L ∂ y ) d A . the statement of Green’s theorem on p. 381). Green's theorem can be applied because C is a closed curve, but the theorem is applicable for positively oriented curves.. It allows us to find the relationship between the line integral and double integral – this is why Green’s theorem is one of the four core concepts of the fundamental theorem of Calculus. In plane geometry, it is used to find the area and centroid of plane figures. To state Green’s Theorem, we need the following def-inition. According to the Green's theorem, when we travel around a closed curve in the same direction, the boundaries of that cure are positive. The path traversal in calculating the Green’s theorem is A. Clockwise B. Anticlockwise C. Inwards D. Outwards Answer: B The cut leads to a counter-clockwise orientation on the outer ring and a clockwise orientation on the inner ring. Transforming to polar coordinates, we obtain. Share. 115. Otherwise we say it has a negative orientation. }\) Solution Let’s first sketch C and D for this case to make sure that the conditions of Green’s Theorem are met for C and will need the sketch of D to evaluate the double integral. Notice that the standard orientation implies that the outer curve traverses counterclockwise whereas the inner curve goes in clockwise direction. Let Dbe the region between the two curves. b ) Anticlockwise. Given that, F = ( x − y) i + ( y − x) j. 17-4 Greens Theorem.pdf - Section 17.4: Green’s Theorem... School University of North Carolina, Chapel Hill; Course Title MATH 233; Since D D is a disk it seems like the best way to do this integral is to use polar coordinates. However, we know that if we let x be a clockwise parametrization of Cand y an Using Green's Theorem, compute the counterclockwise circulation of $\mathbf F$ around the closed curve C. $$\mathbf F = (-y - e^y \cos x)\mathbf i + (y - … If this cut becomes so thin as to vanish, then the line integrals along the lines introducing the cut will cancel off and we have a boundary consisting of two curves with opposite orientations. We want to compute the line integral of F = yi+xj x2+y2 over the curve Cdrawn above. Method 2 (Green’s theorem). GREEN’S THEOREM Green’s Theorem used to integrate the derivatives in a. Remember that P P is multiplied by x x and Q Q is multiplied by y y. So in our example, where we're going clockwise, the region is to our right, Green's theorem is going to be the negative of this. Green’s Theorem: LetC beasimple,closed,positively-orienteddiﬀerentiablecurveinR2,and letD betheregioninsideC. Calculate the Green’s value for the functions F = y 2 and G = x 2 for the region x = 1 and y = 2 from origin . Thursday,November10 ⁄⁄ Green’sTheorem Green’s Theorem is a 2-dimensional version of the Fundamental Theorem of Calculus: it relates the (integral of) a vector ﬁeld F on the boundary of a region D to the integral of a suitable derivative of F over the whole of D. 1.Let D be the unit square with vertices (0,0), (1,0), (0,1), and (1,1) and consider the vector ﬁeld The given function is, F ( x, y) = ( x + 4 y 3, 4 x 2 + y) Step 2. So in this situation when the region is to our right and we're going-- so this is counterclockwise. dr. (Check the orientation of the curve before applying the theorem.) 3. (You proved half of the theorem in a homework assignment.) Solution: curl(F) = 2, so that G 2dA= 2Area(G) = 578. Line Integrals and Green’s Theorem Jeremy Orlo 1 Vector Fields (or vector valued functions) Vector notation. Uses of Green's Theorem . Let C 1 be the unit circle oriented clockwise and consider the combined curve C+ C 1. Note that as the circle on the integral implies the curve is in the positive direction and so we can use Green’s Theorem on this integral. Green’s Theorem in two dimensions can be interpreted in two diﬀerent ways, both leading to important generalizations, namely Stokes’s Theorem and the Divergence ... curve is counter-clockwise. Green’s Theorem in two dimensions can be interpreted in two diﬀerent ways, both leading to important generalizations, namely Stokes’s Theorem and the Divergence ... curve is counter-clockwise. Also, it is used to calculate the area; the tangent vector to the boundary is rotated 90° in a clockwise direction to become the outward-pointing normal vector to derive Green’s Theorem’s divergence form. dr. (Check the orientation of the curve before applying the theorem.) where is the circle with radius centered at the origin. Reply. You da real mvps! Example 2. In this blog post, I will prove that a very elegant theorem for the area of a simple polygon based on Green’s theorem is true. 1 so that Green’s Theorem applies, which we do in the example below: Example 2. We write the components of the vector fields and their partial derivatives: Then. Green’s Theorem (Curl Form) MTH 261 Calculus III Delta College Green’s Theorem (Curl Form) The counter-clockwise circulation of a eld F~(x;y) = hf(x;y);g(x;y)iaround a simple, closed curve C is equal to the double integral of the curl of F~ over the region R enclosed by C. I C F~ T ds~ = Z Z R curl F dA~ I C f dx+ gdy = Z Z R @g @x @f @y! Writing − C 1 means "traverse C 1 in the opposite direction". ∮Cf dy−g dx , where f,g=8x2,8y2 and C is the upper half of the unit circle and the line segment −1≤x≤1 oriented clockwise. 21:17 integral of curl over this region here, the unit disk. It is important to note the parameter of the curve. Green’s theorem 7 Then we apply (⁄) to R1 and R2 and add the results, noting the cancellation of the integrationstaken along the cuts. Figure 3. If you are integrating clockwise around a curve and wish to apply Green's theorem, you must flip the sign of your result at some point. Green’s theorem takes this idea and extends it to calculating double integrals. Let S S be an oriented smooth surface that is bounded by a simple, closed, smooth boundary curve C C with positive orientation. We’ll also discuss a ... and (3;4), oriented clockwise. By Green’s Theorem, F conservative ()0 = I C Pdx +Qdy = ZZ De ¶Q ¶x ¶P ¶y dA for all such curves C. This says that RR De ¶Q ¶x ¶ P ¶y dA = 0 independent of the domain De. dA Report Save Follow. Who are the experts? We review their content and use your feedback to keep the quality high. More posts from the learnmath community. Step 2. It is very important to learn how to use the Green’s Theorem precisely. (Recall that, in Green's theorem, when you walk along a boundary curve in the direction of the arrow, $R_a$ has to be on your left.). Example 1 Use Green’s Theorem to evaluate where C is the triangle with vertices, , with positive orientation. ∮ C F ⋅ d r = − ∫ ∫ D ( ∂ Q ∂ x − ∂ P ∂ y) d A. Orientation in Green’s Theorem. (1) where the left side is a line integral and the right side is a surface integral. De nition. x16.4 Green’s Theorem 1. Let . We can travel clockwise or anti-clockwise. N d s and − S − S is oriented counterclockwise. Use Green's Theorem to evaluate the following line integral. 0 pdV for a curve which goes clockwise around a closed region R. Green’s theorem tells us that the work done is the area of the region R. ELECTROMAGNETISM. Let's write P(x, … Example 1 ∮ ⃗-where . Greens Theorem Green’s Theorem gives us a way to transform a line integral into a double integral. Thanks to all of you who support me on Patreon. Green’s Theorem: Recall that if F = hP, Qiis conservative, then Z C Fdr = 0 for any piecewise smooth closed curve C. Green’s theorem helps us to calculate Z C Fdr = Z C Pdx +Qdy of general (not conservative) vector ﬁeld F along a closed curve C via double integral over the domain D bounded by C. The path traversal in calculating the Green’s theorem is a) Clockwise b) Anticlockwise c) Inwards d) Outwards 2 2 . . Outward flux: ∮ C F. n d s = ∫ … Our standing hypotheses are that γ : [a,b] → R2 is a piecewise It is very important to learn how to use the Green’s Theorem precisely. 17-4 Greens Theorem.pdf - Section 17.4: Green’s Theorem Green’s Theorem: Let C be a simple closed piecewise smooth curve, oriented counter clockwise. What Is Green’s Theorem? Using Green's Theorem to Calculate the Counter-Clockwise Circulation for the Field F and Curve C. Use Green’s Theorem to find the counter-clockwise circulation for the field F and curve C. Green's Theorem says that the counter-clockwise circulation is ∮ C F ⋅ T d s = ∮ C M d x + N d y. I will use the latter formula. from (0,0) to (1,1), and the upper-left bounds. counter clockwise. IfF(x;y) = 2 4 P(x;y) Q(x;y) 3 5isacontinuouslydiﬀerentiablevectorﬁeld deﬁnedonD,then: I C Fdr = ZZ D (r F)kdA Whilethis“vector” versionofGreen’sTheoremisperhapsmorediﬃculttousecomputationally,itiseasier … Green’s theorem allows us to integrate regions that are formed by a combination of a line and a plane. Green’s theorem con rms that this is the area of the region below the graph. Evaluate. Let S be the region in the first quadrant of R2 bounded by the curve y = 3 − x2 + 2x , and compute ∫∂S(xy + sin(ey))dx + xeycos(ey)dy. Green’s Theorem Example Evaluate R C xydx+ x2dywhere Cis the rectangle with vertices (0;0);(3;0);(3;1);(0;1) oriented counter-clockwise. So, Green's theorem gives us. Since we now know about line integrals and double integrals, we are ready to learn about Green's Theorem. 17-4 Greens Theorem.pdf - Section 17.4: Green’s Theorem... School University of North Carolina, Chapel Hill; Course Title MATH 233; Over a region in the plane with boundary , Green's theorem states. F (x, y) = . This problem may look familiar as it was on the Line Integral \Quiz". The counter clockwise oriented CRRencloses the island Gwhich has 289 unit squares. Theorem 15.4.1 Green’s Theorem Let R be a closed, bounded region of the plane whose boundary C is composed of finitely many smooth curves, let r → ⁢ ( t ) be a counterclockwise parameterization of C , and let F → = M , N where N x and M y are continuous over R . $\frac{\partial B_2}{\partial x}(x,y) - \frac{\partial B_1}{\partial y}(x,y)$ really is zero at all points $(x,y)$ of the region \(R_a\text{. Step 3. F(x, y) = (y – cos(y), x sin(y)), C is the circle (x - 5)2+ (y + 4)² = 9 oriented clockwise. This is only possible if ¶Q ¶x = ¶P ¶y everywhere. In 18.04 we will mostly use the notation (v) = (a;b) for vectors. Deﬁnition 1.1. From the integral we have, P = x y 2 + x 2 Q = 4 x − 1 P = x y 2 + x 2 Q = 4 x − 1. View Answer. Also let →F F → be a vector field then, ∫ C →F ⋅ d→r = ∬ S curl →F ⋅ d→S ∫ C F → ⋅ d r → = ∬ S curl F → ⋅ d S →. One of the 2DMaxwell equationis curl(E) = −1 c d This can also be written compactly in … (Solution)In our symbolic notation, we’re being asked to compute C F dr, where F = hlnx+ y; x2i. Use Green’s theorem to evaluate line integral where C is any smooth simple closed curve joining the origin to itself oriented in the counterclockwise direction. Now, using Green’s theorem on the line integral gives, ∮ C y 3 d x − x 3 d y = ∬ D − 3 x 2 − 3 y 2 d A ∮ C y 3 d x − x 3 d y = ∬ D − 3 x 2 − 3 y 2 d A. where D D is a disk of radius 2 centered at the origin. Clarification: In physics, Green’s theorem is used to find the two dimensional flow integrals. When David took out some blue and sticks and replaced them with an equal number of green sticks, the ratio of the number of blue sticks to the number of green sticks became 3:1. counter clockwise, and C is the boundar y of a region . 17-4 Greens Theorem.pdf - Section 17.4: Green’s Theorem Green’s Theorem: Let C be a simple closed piecewise smooth curve, oriented counter clockwise. These theorems also fit on this sort of diagram: The Fundamental Theorem of Line Integrals is in some sense about “undoing” the gradient. F (x, y) = e3x + x2y, e3y − xy2 C is the circle x2 + y2 = 25 oriented clockwise. By Green’s theorem, the flux is Green’s theorem 7 Then we apply (⁄) to R1 and R2 and add the results, noting the cancellation of the integrationstaken along the cuts. Green’s Theorem can be used to prove important theorems such as $2$-dimensional case of the Brouwer Fixed Point Theorem (in Problem Set 8). Show Step 2. Use Green’s Theorem to evaluate ∫ C (y4−2y) dx−(6x−4xy3) dy ∫ C ( y 4 − 2 y) d x − ( 6 x − 4 x y 3) d y where C C is shown below. ∬ D N x − M y d x d y = ∮ ( − C 1) ∪ C 2 M d x + N d y = ∮ − C 1 M d x + N d y + ∮ C 2 M d x + N d y = − ∮ C 1 M d x + N d y + ∮ C 2 M d x + N d y. Compute the line integral , where is the unit circle, oriented clockwise. Green’s theorem takes this idea and extends it to calculating double integrals. As rotations in two dimensions are determined by a single angle, in three dimensions, three parameters are needed. 21:21 And, of course the curl is zero, well, except at the. For a C1 vector eld F = [P;Q] in a region GˆR2, the curl is de ned as curl(F) = Q x P y. Method 2 (Green’s theorem). According to the Green's theorem, when we travel around a closed curve in the same direction, the boundaries of that cure are positive. **This is clearly a very weird line integral. This is illustrated by the following plot. Solution. and gives: C: boundary of the region lying between … View LESSON 11 - Green's theorem.pdf from SCHOOL OF 12563 at University of Cagayan Valley (Cagayan Colleges Tuguegarao). The clockwise orientation of the boundary of a disk is a negative orientation, for example. Since all three conditions are satisfied, we can use Green’s theorem. However, we know that if we let x be a clockwise parametrization of Cand y an Use $-C_a$ to denote $C_a$ oriented clockwise. Since we now know about line integrals and double integrals, we are ready to learn about Green's Theorem. At the origin, 21:27 the vector field is not defined. C is the boundary ofIf , , , and P Q P Qyx a region .R xy RC ³³ ³Q P dA Pdx Qdy are continuous in , then:R We can use Green's theorem to compute ar eas: 1 2 R R R area R xdy ydx xdy ydx w w w ³ ³ ³ If in the region between and we have12: , and , , , are continuous, then we can use Green's Theorem: x y y x CC Fill the box with correct option please and I will give you Thumbs up. Find step-by-step Calculus solutions and your answer to the following textbook question: Use Green’s Theorem to evaluate the line integral along the given positively oriented curve. Transcribed Image Text: F (x, y) = 5x² yi-2x j Use Green's Theorem to calculate the work done by the force field on a particle that moves around the curve C bounded by y=-x² +10 and y=-4x+13 in the counter-clockwise direction. The counter clockwise oriented CRRencloses the island Gwhich has 289 unit squares. ⃗ =〈! The clockwise orientation of the boundary of a disk is a negative orientation, for example. An engineering application is the planimeter, a mechanical device for mea- ... y12 + cos(y) + 3x] counter clockwise along the boundary in terms of A. A convenient way of expressing this result is to say that (⁄) holds, where the orientation {\displaystyle \oint _{C}(M,-L)\cdot \mathbf {\hat {n}} \,ds=\iint _{D}\left(\nabla \cdot (M,-L)\right)\,dA=\iint _{D}\left({\frac {\partial M}{\partial x}}-{\frac {\partial L}{\partial y}}\right)\,dA.} We can travel clockwise or anti-clockwise. It reads: Use green's theorem to calculate the work done by the force F on a particle that is moving counterclockwise around the closed path C. It gives: F (x,y)= (3x^2+y)i + 4xy^2j. But with simpler forms. 1. Therefore, the answer is: . C is a closed curve and using Green’s theorem for clockwise orientation the integral is evaluated using the below formula. Note that the clockwise orientation on C 1 is compatible (i.e. Warning: Green's theorem only applies to curves that are oriented counterclockwise. Note that Green's Theorem applies to regions in the xy-plane. We can use Green's. 21:25 origin. Because the path Cis oriented clockwise, we cannot im-mediately apply Green’s theorem, as the region bounded by the path appears on the right-hand side as we traverse the path C(cf. This is illustrated by the following plot. 21:13 the line integral along this loop is equal to the double. It is negative for clockwise paths. Question. 1. The result still is (⁄), but with an interesting distinction: the line integralalong the inner portion of bdR actually goes in the clockwise direction. The line integrals along the four sides are as follows: • The right side: Z y 0+∆y y 0 Q(x 0 +∆x,y)dy; (3.1) • The top: Z x 0 x 0+∆x If there were 185 green sticks in the box now, (a) find the total number of blue and green sticks in the box, (b) find the number of green sticks in the box at first. Particularly in a vector field in the plane. The other common notation (v) = ai + bj runs the risk of i being confused with i = p 1 {especially if I forget to make i boldfaced. Calculus III - Green's Theorem (Practice Problems) Use Green’s Theorem to evaluate ∫ C yx2dx−x2dy ∫ C y x 2 d x − x 2 d y where C C is shown below. $1 per month helps!! Let’s work a couple of examples. C is the circle x 2 + y 2 = 16 oriented clockwise. F(x, y) = e2x + x2y, e2y − … Green's theorem (circulation form): ∮ C F. d r = ∫ ∫ R ( ∂ N ∂ x − ∂ M ∂ y) d x d y. Real line integrals. Question: Use Green's Theorem to evaluate the following line integral. Green’s Theorem is the particular case of Stokes Theorem in which the surface lies entirely in the plane. 4. Calculating Areas A powerful application of Green’s Theorem is to ﬁnd the area inside a curve: Theorem. Orientation in Green’s Theorem. Thursday,November10 ⁄⁄ Green’sTheorem Green’s Theorem is a 2-dimensional version of the Fundamental Theorem of Calculus: it relates the (integral of) a vector ﬁeld F on the boundary of a region D to the integral of a suitable derivative of F over the whole of D. 1.Let D be the unit square with vertices (0,0), (1,0), (0,1), and (1,1) and consider the vector ﬁeld A convenient way of expressing this result is to say that (⁄) holds, where the orientation Green’s theorem relates the work done by a vector eld on the boundary of a region in R2 to the integral of the curl of the vector eld across that region. … Example 3. Stokes’ Theorem. Unit 31: Green’s theorem Lecture 31.1. Applications of Green’s theorem are meant to be in (a) One dimensional (b) Two dimensional (c) Three dimensional (d) four dimensional ANSWER: _____ 11. Solution: curl(F) = 2, so that G 2dA= 2Area(G) = 578. the statement of Green’s theorem on p. 381). Because the path Cis oriented clockwise, we cannot im-mediately apply Green’s theorem, as the region bounded by the path appears on the right-hand side as we traverse the path C(cf. Assume the curve is oriented counterclockwise. 56-57): If γ is a simple closed curve in the plane, then the Green's Theorem implies that ∫∂Sxdy = − ∫∂Sydx = ∫∂S1 2(xdy − ydx) = ∬S1dA = area(S). Question. Now apply the “serious application” of Green’s Theorem proved in the last section to g, with D\{p} playing the role of “the open set containing Ω and Γ.” The result is: 0 = Z Γ g(z)dz = γ−γε g(z)dz = γ g(z)dz − γε g(z)dz, 3Recall the Jordan Curve Theorem (pp. By Green’s theorem, the curl evaluated at (x,y) is limr→0 R Cr F dr/~ (πr2) where C r is a small circle of radius r oriented counter clockwise an centered at (x,y). Twice the area of . However, we also have our two new fundamental theorems of calculus: The Fundamental Theorem of Line Integrals (FTLI), and Green’s Theorem. The line integrals along the four sides are as follows: • The right side: Z y 0+∆y y 0 Q(x 0 +∆x,y)dy; (3.1) • The top: Z x 0 x 0+∆x Just show that the … Green’s Theorem, Cauchy’s Theorem, Cauchy’s Formula These notes supplement the discussion of real line integrals and Green’s Theorem presented in §1.6 of our text, and they discuss applications to Cauchy’s Theorem and Cauchy’s Formula (§2.3). Use Green's Theorem to evaluate F. dr. (Check the orientation of the curve before applying the theorem.) Use Green's Theorem to evaluate the following line integral. $ (4y + 7,4x² − 2) • dr, where C is the boundary of the rectangle with vertices (0,0), (6,0), (6,4), and (0,4) C $ (4y + 7,4x² - 2) • dr = (Type an exact answer.) It can also be used to complete the proof of the 2-dimensional change of variables theorem, something we did not do. $$ ∫c (y + e^√x)dx+(2x+cosy^2)dy, $$ C is the boundary of the region enclosed by the parabolas $$ y … Green's Theorem can be used to prove important theorems such as $2$-dimensional case of the Brouwer Fixed Point Theorem. The result still is (⁄), but with an interesting distinction: the line integralalong the inner portion of bdR actually goes in the clockwise direction. Green’s theorem explains so what the curl is. Step 1: The vector field is .. Green's theorem : If C be a positively oriented closed curve, and R be the region bounded by C, M and N are the partial derivatives on an open region then. Use Green's Theorem to evaluate F. dr. (Check the orientation of the curve before applying the theorem.) Solution. In the plane (ﬂatland), the electric ﬁeld is a vector ﬁeld E = (E1,E2), while the magnetic ﬁeld is a scalarﬁeld. Theorem 18.4.1 (Green's Theorem) If the vector field and the region are sufficiently nice, and if is the boundary of ( is a closed curve), then provided the integration on the right is done counter-clockwise around . Using Greens theorem. Green's theorem-- this applies when the region is to our left. A Quick Application Green's theorem holds for any vector field, so long as C is closed! It is important to note the parameter of the curve. To indicate that an integral is being done over a closed curve in the counter-clockwise direction, we usually write . Green’s theorem also applies when the region is no simply connected, that is, it has holes. Green's Theorem says: for C a simple closed curve in the xy -plane and D the region it encloses, if F = P ( x, y ) i + Q ( x, y ) j, then where C is taken to have positive orientation (it is traversed in a counter-clockwise direction). This curve rotates in a clockwise direction, so negative should … Theorem 16.4.1 (Green's Theorem) If the vector field F = P, Q and the region D are sufficiently nice, and if C is the boundary of D ( C is a closed curve), then. 21.17. Green's theorem would tell me. Use Green’s theorem to evaluate line integral where C is circle oriented in the clockwise direction. By Green's theorem, if C is the circle z+y =D4 taken anticlockwise, then 'dy, equals Question Transcribed Image Text: By Green's theorem, if C is the circle r + y = 4 taken anticlockwise, then equals 27. none of these options. Needed to be solved correclty in 1 hour completely and get the thumbs up please show neat and clean work.box answer should be right. Using Green's theorem, calculate the integral The curve is the circle (Figure ), traversed in the counterclockwise direction. Solution. F(x,y)=, C is the triangle from (0, 0) to (0, 4) to (2, 0) to (0, 0). Figure 1. We say a closed curve C has positive orientation if it is traversed counterclockwise. A sketch is helpful. The Similarity Green’s Theorem Stokes’ Theorem Both relate closed line integrals with surface integrals. That's what Green's theorem says: integrate the curl (the amount of microscopic counter-clockwise twisting force) to get the counter-clockwise work done around the boundary. Step 1. Please note that Green’s theorem is a standard theorem from multi variable calculus. If , , , and are continuous in , then yx C R P Q P Q R Green's Theorem xy RC ³³ ³Q P dA Pdx Qdy We can use Green's theorem to compute ar eas: 1 2 R R R area R xdy ydx xdy ydx w w w Review: ³ ³ ³: (a) If is defined in a connected a nd simply connected region, Use green's theorem to calculate the integral Fdr, F(r, y) = (e* + x²y , e³ – ry² + sen(3) + 2) where knowing that e cis the path found when traveling clockwise the boundary of a circle of radius 6 centered on the origin Use Green’s Theorem to evaluate ∫ C (6y −9x)dy −(yx −x3) dx ∫ C ( 6 y − 9 x) d y − ( y x − x 3) d x where C C is shown below. Unit 31: Green’s theorem Lecture 31.1. Green’s … Figure 3. If is the closed region enclosed by , Green’s theorem says This integral is equal to: Twice the length of . Solution. Where, F = ( P, Q) = ( x + 4 y 3, 4 x 2 + y) Step 3. F(x, y) = (y – cos(y), x sin(y)), C is the circle (x – 5)2+ (y + 4)² = 9 oriented clockwise. For a C1 vector eld F = [P;Q] in a region GˆR2, the curl is de ned as curl(F) = Q x P y. Green's theorem is a vector identity which is equivalent to the curl theorem in the plane. !, and C is the parabola=! Sketch the curve C. 92 3 ans: Experts are tested by Chegg as specialists in their subject area. dr. (Check the orientation of the curve before applying the theorem.) :) https://www.patreon.com/patrickjmt !! Find step-by-step solutions and your answer to the following textbook question: Use Green’s Theorem to evaluate integral C F.dx (Check the orientation of the curve before applying the theorem.)

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