Question

2. Consider the line integral I C F · d r, where the vector field F = x(cos(x 2 ) + y)i + 2y 3 (e y sin3 y + x 3/2 )j and C is the closed curve in the first quadrant consisting of the curve y = 1 − x 3 and the coordinate axes x = 0 and y = 0, taken anticlockwise.

(a) Use Green’s theorem to express the line integral in terms of a double integral over the region R enclosed by C.

(b) Calculate the double integral using a repeated integral with respect to y first and x second.

(c) Now reverse the order of integration and evaluate the double integral using a repeated integral with respect to x first and y second.

Answer #1

Use Divergence theorem to evaluate surface integral S F ·n dA
where S is the surface of the solid enclosed by the tetrahedron
formed by the coordinate planes x = 0, y = 0 and z = 0 and the
plane 2x + 2y + z = 6 and F = 2x i − x^2 j + (z − 2x + 2y) k.

For each vector field F~ (x, y) = hP(x, y), Q(x, y)i, find a
function f(x, y) such that F~ (x, y) = ∇f(x, y) = h ∂f ∂x , ∂f ∂y i
by integrating P and Q with respect to the appropriate variables
and combining answers. Then use that potential function to directly
calculate the given line integral (via the Fundamental Theorem of
Line Integrals):
a) F~ 1(x, y) = hx 2 , y2 i Z C F~ 1...

Use Green’s theorem to evaluate the integral: ∫(-x^2y)dx
+(xy^2)dy where C is the boundary of the region enclosed by y=
sqrt(9 − x^2) and the x-axis, traversed in the counterclockwise
direction.

Calculate double integral D f(x, y) dA as an iterated integral,
where f(x, y) = −4x 2y 3 + 4y and D is the region bounded by y = −x
− 3 and y = 3 − x 2 .

evaluate the double integral where f(x,y) = 6x^3*y - 4y^2 and D
is the region bounded by the curve y = -x^2 and the line x + y =
-2

using the change of variable x =u/v, y=v evaluate "double
integral(x^2+2y^2)dxdy: R is the region in the first quadrant
bounded by the graphs of xy=1, xy=2, y=x, y=2x

(1 point) Evaluate the line integral ∫F⋅d r∫CF⋅d r where
F=〈-5sinx,-2cosy,10xz〉 and C is the path given by
r(t)=(2t^3,-3t^2,-2t) for 0≤t≤10≤t≤1
∫F⋅d r=

Evaluate the line integral ∫F⋅d
r∫CF⋅d r where
F=〈sinx,−3cosy,5xz〉 and C is the path given by
r(t)=(-2t^3,-3t^2,3t) for 0≤t≤1

Consider the vector
field F = ( 2 x e y − 3 ) i + ( x 2 e y + 2 y ) j ,
(a) Find all potential
functions f such that F = ∇ f .
(b) Use (a) to
evaluate ∫ C F ⋅ d r , where C is the curve r ( t ) = 〈 t , t 2 〉 ,
1 ≤ t ≤ 2 .

57.
a. Use polar coordinates to compute the (double integral (sub
R)?? x dA, R x2 + y2) where R is the region in the first quadrant
between the circles x2 + y2 = 1 and x2 + y2 = 2.
b. Set up but do not evaluate a double integral for the mass of
the lamina D={(x,y):1≤x≤3, 1≤y≤x3} with density function ρ(x, y) =
1 + x2 + y2.
c. Compute??? the (triple integral of ez/ydV), where E=
{(x,y,z):...

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