AP Calc AB Problem - Finding volumeThe base of a solid in the region bounded...

detineerlf

detineerlf

Answered

2022-07-16

AP Calc AB Problem - Finding volume
The base of a solid in the region bounded by the two parabolas y 2 = 8 x and x 2 = 8 y. Cross sections of the solid perpendicular to the x-axis are semicircles. What is the volume, in cubic units, of the solid?
The answer choices are:
A) 288 π 35
B) 576 π 35
C) 144 π 35
D) 8 π
I started off by writing the integral like this:
0 8 1 2 π ( 1 2 ( 8 x x 2 8 ) ) 2 d x
then I simplified it to π 8 0 8 ( 8 x x 2 8 ) 2 d x
I can't think of anyway to solve this problem from here, without using a calculator.

Answer & Explanation

slapadabassyc

slapadabassyc

Expert

2022-07-17Added 21 answers

Step 1
Expand the binomial term:
V = π 8 x = 0 8 ( 8 x x 2 8 ) 2 d x = π 8 x = 0 8 8 x 2 8 x x 2 8 + x 4 64 d x .
Step 2
Now integrate term by term:
V = π 8 [ 4 x 2 2 7 x 7 / 2 + x 5 5 ( 64 ) ] x = 0 8 .

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