Question

2. Evaluate the double integral Z Z R e ^(x^ 2+y ^2) dA where R is the semicircular region bounded by x ≥ 0 and x^2 + y^2 ≤ 4.

3. Find the volume of the region that is bounded above by the sphere x^2 + y^2 + z^2 = 2 and below by the paraboloid z = x^2 + y^2 .

4. Evaluate the integral Z Z R (12x^ 2 )(y^3) dA, where R is the triangle with vertices (0, 0), (1, 1) and (2, 0).

ZZ means double integral. All x's are variables. Thank you!

Answer #1

1. Let R be the rectangle in the xy-plane bounded by the lines x
= 1, x = 4, y = −1, and y = 2. Evaluate Z Z R sin(πx + πy) dA.
2. Let T be the triangle with vertices (0, 0), (0, 2), and (1,
0). Evaluate the integral Z Z T xy^2 dA
ZZ means double integral. All x's are variables. Thank you!.

Use the given transformation to evaluate the double integral.
(12x + 12y) dA R , where R is the parallelogram with vertices (−3,
6), (3, −6), (4, −5), and (−2, 7) ; x = 1/ 3 *(u + v), y = 1 /3* (v
− 2u)

Use the given transformation to evaluate the double integral of
(x-6y) dA, where R is the triangular region with vertices
(0, 0), (5, 1), and (1, 5).
x = 5u +
v, y = u +
5v

Use a change of variables to evaluate Z Z R (y − x) dA, where R
is the region bounded by the lines y = 2x, y = 3x, y = x + 1, and y
= x + 2. Use the change of variables u = y x and v = y − x.

Use the given transformation to evaluate the integral.
(12x + 12y) dA
R
, where R is the parallelogram with vertices
(−2, 4),
(2, −4),
(5, −1),
and
(1, 7)
; x =
1
3
(u + v), y =
1
3
(v − 2u)

Consider the integral ∫∫R(x^2+sin(y))dA where R is the region
bounded by the curves x=y^2, x=4, and y=0. Setup up this
integral.

Use the given transformation to evaluate the integral.
(x −
8y) dA,
R
where R is the triangular region with vertices (0, 0),
(7, 1), and (1, 7).
x = 7u +
v, y = u +
7v

Evaluate the double integral of 5x3cos(y3)
dA where D is the region bounded by y=2, y=(1/4)x2, and
the y-axis.

Use the given transformation to evaluate the integral.
6xy dA
R
, where R is the region in the first quadrant bounded
by the lines y =
2
3
x and y =
3
2
x and the hyperbolas xy =
2
3
and xy =
3
2
;
x = u/v, y = v

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 .

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