Question

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): 0≤y≤1, y≤x≤1, 0≤z≤xy}.

Answer #1

Evaluate the given integral by changing to polar
coordinates.
R
(5x − y) dA, where R is the region in the first
quadrant enclosed by the circle
x2 + y2 = 16
and the lines
x = 0
and
y = x

Use a double integral in polar coordinates to find the volume of
the solid bounded by the graphs of the equations.
z = xy2, x2 + y2 =
25, x>0, y>0, z>0

Evaluate the double integral ∬Ry2x2+y2dA, where R is the region
that lies between the circles x2+y2=9 and x2+y2=64, by changing to
polar coordinates .

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...

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 =
1
2
x and y =
3
2
x and the hyperbolas xy =
1
2
and xy =
3
2
;
x = u/v, y = v

Set-up, but do not evaluate, an iterated integral in polar
coordinates for ∬ 2x + y dA where R is the region in the xy-plane
bounded by y = −x, y = (1 /√ 3) x and x^2 + y^2 = 3x. Include a
labeled, shaded, sketch of R in your work.

use a double integral in polar coordinates to find the volume of
the solid in the first octant enclosed by the ellipsoid
9x^2+9y^2+4z^2=36 and the planes x=sqrt3 y, x=0, z=0

Use cylindrical coordinates.
Evaluate the integral, where E is enclosed by the
paraboloid
z = 8 + x2 + y2,
the cylinder
x2 + y2 = 8,
and the xy-plane.
ez dV
E

Use cylindrical coordinates.
Evaluate the integral, where E is enclosed by the
paraboloid
z = 7 + x2 + y2,
the cylinder
x2 + y2 = 8,
and the xy-plane.
ez dV
E

Use polar coordinates and double integrals to compute the
improper integral: sqrt{x} (e^{-x) dx.

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