Question

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

Answer #1

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

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.

Sketch the region R bounded by the graph of y=1-(x+2)^2 and the
x-axis. Set up, but do not evaluate, definite integrals that can be
used to find the volumes of the solids obtained by revolving R
about the given lines. In each case state whether your solution
primarily uses the method of disks, washers, or shells.
a) y=0
b)x=0
c)x=4
d)y=-5

Consider the region bounded by the line y = 2x and the parabola
y = 2x2-2x.
a. Evaluate the volume obtained by rotating this region about
the line x = -5
b. Evaluate the volume obtained by rotating this region about
the line y = -10

1. (a) Use the graph paper provided to sketch the region bounded
by the x-axis and the lines x + y = 4 and y = 3x.
(b) Shade the region you just drew above.
(c) Suppose that you were going to use Calculus to compute the
area of the shaded region in part (a) above. If you chose the
x-axis as your axis of integration, then how many integrals would
be needed to compute this area?
(d) Suppose that...

Consider the region bounded by ? = 4? , ? = 1 and x-axis. Set
up the appropriate integrals
for finding the volumes of revolution using the specified
method and rotating about the specified axis. Be sure to first
sketch the region and draw a typical cross section. SET UP THE
INTEGRALS ONLY. DO NOT evaluate the integral.
a) Disc/washer method about the x-axis
b) Shell method about the y-axis
c) Disc/washer method about the line ? = 2.
d)...

Consider the region R bounded by y = sinx, y = −sinx , from x =
0, to x=π/2.
(1) Set up the integral for the volume of the solid obtained by
revolving the region R around
x = −π/2
(a) Using the disk/washer method.
(b) Using the shell method.
(2) Find the volume by evaluating one of these integrals.

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.

Consider the set D which is the triangular region bounded by
y=x, x=0, y=4 and the
boundary of this triangular region. Find the absolute maximum
and minimum values of f(x,y) = x2 -xy
+y2+1
on D. Make sure you show work for testing for critical values
inside the region and on the lines that make up the boundary as
part of your work shown

let R be the region bounded by the curves x = y^2 and x=2y-y^2.
sketch the region R and express the area R as an iterated integral.
(do not need to evaluate integral)

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