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Find the volume of the described solid $S .$A frustum of a pyramid with square base of side $b,$ squaretop of side $a,$ and height $h$What happens if $a=b ?$ What happens if $a=0 ?$

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$V=\frac{1}{3} h\left(a^{2}+a b+b^{2}\right)$If $a=b$ then it is a rectangular boxIf $a=0$ then it is a pyramid

Calculus 2 / BC

Chapter 7

APPLICATIONS OF INTEGRATION

Section 2

Volumes

Applications of Integration

Campbell University

Harvey Mudd College

Baylor University

Lectures

07:42

Find the volume of the des…

0:00

05:14

A frustum of a pyramid wit…

09:21

$47-59$ Find the volume of…

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01:19

05:22

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07:40

A frustum of a pyramid is …

for this problem, we want to find the volume of the solid. Um So we start with the side view of the frost um uh and were given the different values so we know that this side A B. Is equal to be decided C. D. Is equal to a. Um And we're given these cross sections. So the first thing we want to do is find the equation of the line that passes through B. C. So using our point slope form will have one minus y. One equals M times x minus X one. Then I'm looking at the first um what we end up getting for our line is going to be that Y equals a minus B over to H X plus B over to. Then we have a square ass of acts and we see that that is going to equal two times A minus B over to H X plus B over to which ends up giving us a minus B over H X plus B. Then we have a square to get the area of one cross section. So that's gonna be V equals the integral from zero to H. Of S squared dx. We integrate this and what we end up getting is one third one third H A squared plus A B plus B squared. Then if we let a equal B we end up getting that. The volume is 1/3, H. A squared plus a squared plus A squared. That's three A squared. So that gets rid of this leaving us with H. A. Squared. And then it is a rectangular box. So it would be a cube. If H equaled A. Which equals B. If A equals zero then what we end up seeing is that V will be one third H. B squared. And this matches the pyramid volume formula with a square base. So those are the different scenarios and those are the values that we end up getting.

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