MAT-1120 Midterm: MATH 1120 App State Spring2009 Test2 answer key

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Math 1120 Test #2 — ANSWER KEY March 3rd, 2009

1. (16 points) Volumes.

(a) Find an integral which computes the volume of the solid obtained by rotating the region bounded by y=√x,

y= 0, and x= 4 about the axis x=−3y=−3.

Do not evaluate your integral.

Volume = Z4

π(√x−(−3))2−π(0 −(−3))2dx =Z4

π(3 + √x)2−9dx

[If we wanted to compute the answer... = πR4

09 + 6√x+x−9dx =π4x3/2+ (1/2)x2



0= 40π.]

Note: The problem originally said to rotate about x=−3. In this case, we would need the following integral to

compute volume:

Volume = Z2

π(4 −(−3))2−π(y2−(−3))2dy =Z2

49π−π(y2+ 3)2dy =288

5π

(b) The Great Pyramid of Giza has a square base which is about 750 feet long on each side. It is currently about 450

feet tall (it continues to shrink because of erosion). Find an integral which computes the volume the pyramid.

Do not evaluate your integral.

“Slice” the pyramid into rectangular solids with square bases. Let xbe the distance (in feet) from the top of the

pyramid. Also, let sbe the length (in feet) a side of a slice which is xfeet from the top of the pyramid.

We know that when x= 0, s= 0 (at the top the pyramid has width 0). When x= 450 (the bottom of the

pyramid) we have s= 750. The sides of the pyramid are ﬂat, so sand xare linearly related. The slope of this line

is m= (750 −0)/(450 −0) = 750/450 = 5/3. Therefore, s−0 = (5/3)(x−0). Thus s= (5/3)x. So the volume

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( x ( 3))2 (0 ( 3))2 dx = z 4. [if we wanted to compute the answer = r 4. Note: the problem originally said to rotate about x = 3. In this case, we would need the following integral to compute volume: (4 ( 3))2 (y2 ( 3))2 dy = z 2. 5 (b) the great pyramid of giza has a square base which is about 750 feet long on each side. It is currently about 450 feet tall (it continues to shrink because of erosion). Find an integral which computes the volume the pyramid. Slice the pyramid into rectangular solids with square bases. Let x be the distance (in feet) from the top of the pyramid. Also, let s be the length (in feet) a side of a slice which is x feet from the top of the pyramid. We know that when x = 0, s = 0 (at the top the pyramid has width 0).

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Change the order of integration and evaluate the triple integral... integral^(pi)_0 integral^(pi)_0 integral^5sin(phi)_0 (rho)^2*sin(phi)d(rho)d(theta)d(phi) please show all work and be sure to change the order of integration... Question #6

number 6

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MAT-1120 Midterm: MATH 1120 App State Spring2009 Test2 answer key

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