Let D be the region in the xy-plane bounded by y = x and y = x², and C be the associated boundary curve with counter clockwise orientation. (a) Find the intersections of y = x and y = x² and thus sketch the region D. (b) Compute the line integral ${(xy + y²) dx + x² dy} directly by parametrising the path C. (c) Use Green's Theorem in the plane to compute the above line integral by evaluating a double integral.

Q: Evaluate the line integral fx²dx + dy+ydz where C' is the curve r(t) = (t, 2t, t²) for 0 ≤t≤ 3 and…

A: To find :-The line integral  where C is the curve  for

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Q: Evaluate the line integral I of the one-form w = (y+ z) dx + (x + z) dy + 10z dz over the line…

A: We have to find the line integral.

Q: Use Green's Theorem to evaluate the line integral § 2y² dx + 3x² dy, where C is the boundary of the…

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Q: x² + (b) Let C be the curve defined by the intersection of the paraboloid z = y² and the plane y =…

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Q: c) Use Green's theorem to evaluate the line integral x2y2 dx + 4x dy, where C is the boundary of the…

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Q: (x² + xy) dx + (y² - xy) dy; C: the line y = x from the origin to the point (2, 2).

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Q: Suppose a lake is modelled by the region y- x > 0 in the plane. The shore is the straight line y - x…

A: Given: A lake is represented by the region y-x>0 Shore of the lake is the straight line y-x=0…

Q: Use green's theorem to calculate the integral Fdr, F(r, y) = (e* + x²y , e³ – ry² + sen(3) + 2)…

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Q: Calculate line integral F-dr where F(x, y) = (7x + y, 5x + 2y) and C is a rectangle with vertices…

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Q: The line integral y dr x²+ y2'x² + y2 where C is the portion of the unit circle centered at the…

A: nullSolve the line integral∫C-xx2+y2,yx2+y2·drWhere,C is the portion of the unit circle centered at…

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Q: Compute the line integral: y² dx + x² dy two ways. C

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Q: Use Green's theorem to evaluate the line integral r ry dr+r² where C is positively oriented curve…

A: Introduction: The process of determining the area of the region under the curve is known as…

Q: ) Use Green's Theorem to evaluate the line integral along the given positively oriented curve / F.7,…

A: Here we will find out the required value.

Q: Given the line integral | (2x2° + 6y) da + (6x – 2yz) dy + (3a² z² – y²) dz. Let C be the path that…

A: Note that the following integral reduces like this :

Q: Evaluate the line integral I of the one-form w = (y + z) dx + (x + 2) dy + 10z dz over the line…

A: Given: One-form ω=y+zdx+x+zdy+10zdz and the line segment joining 0, 0, 0 and (3, 1, 0). To find:…

Q: Let C be the positively oriented square with vertices (0, 0), (2,0). (2, 2). (0, 2). Use Green's…

A: Green's Theorem:  For a positively oriented curve C∫CP dx+Q dy=∫∫D∂Q∂x-∂P∂y dAfor the region D…

Q: Let the curve C be the line segment from (0, 0) to (3, 1). Let F = ⟨2x-y, 4y-x⟩ Calculate the…

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Q: Evaluate the line integral |(3="y + (1- y):) dz + (z² – z2) dy + z(1 – y) dz where C is any curve…

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Q: TRUE OR FALSE? The given line integral is independant of path.

A: The given line integral is ∫C2xz3+6ydx+6x-2yzdy+3x2z2-y2dz Compare the above line integral with…

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A: Given the surface fx,y,z=-x2+siny-2z2=0. Let us find the equation of the normal to the surface in…

Q: Evaluate the line integral (2x - y + 5)dx + (5y + 3x - 6)dy, for which the path C C traverses around…

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