Question

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.

Answer #1

1a. Using rectangular coordinates, set up iterated integral that
shows the volume of the solid bounded by surfaces z= x^2+y^2+3,
z=0, and x^2+y^2=1
1b. Evaluate iterated integral in 1a by converting to polar
coordinates
1c. Use Lagrange multipliers to minimize f(x,y) = 3x+ y+ 10 with
constraint (x^2)y = 6

Set up iterated integrals for both orders of integration. Then
evaluate the double integral using the easier order.
y dA, D is bounded by y = x
− 20; x = y2
D

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):...

1. (30+30) Set up the double integral in polar coordinates with
the proper limits that represents the volume of the solid bounded
by the paraboloid, ? = 3 − 2?2 − 2?2, and the plane, ? = 1.
Evaluate the integral to find the volume.

Given that D is a region bounded by x = 0, y = 2x, and y =
2.
Given:
∫
∫
x y dA , where D is the region bounded by x = 0, y = 2x, and y =
2.
D
Set up iterated integrals (2 sets) for both orders of
integration. Need not evaluate the
Integrals. Hint: Draw a graph of the region D. Consider D as a
Type 1 or Type 2 region.
Extra credit problem

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 .

Calculate ZZ 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 . SHOW ALL YOUR WORK!

Set up, but do not evaluate, an integral of f(x,y,z) = 20−z over
the solid region deﬁned by
x^2 +y^2 +z^2 ≤ 25 and z ≥ 3. Write the iterated integral(s) to
evaluate this in a coordinate system of your choosing, including
the integrand, order of integration, and bounds on the
integrals.

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

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

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