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Find the volume of the described solid $S .$A pyramid with height $h$ and base an cquilateral trianglewith side $a($ a tetrahedron $)$

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$$V=\frac{\sqrt{3}}{12} a^{2} h$$

Calculus 2 / BC

Chapter 7

APPLICATIONS OF INTEGRATION

Section 2

Volumes

Applications of Integration

Campbell University

University of Nottingham

Boston College

Lectures

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Find the volume of the des…

05:22

03:12

03:33

Find the volume V of the d…

09:02

00:49

02:32

Consider the solid S descr…

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01:19

08:27

$47-59$ Find the volume of…

So we have the different side views of the pyramid, the cross section. So the first thing we want to do is find the equation of the line that passes through A. B. So using point slope form, we end up getting the equation of the line is Y equals A over to H times X. So sfx the side length of the cross section is going to give us two times a over to H. X. Which is just a over a checks, plugging s into the triangle formula. We end up getting a volume equal to the interval from zero to H. Of eight times X. Dx. Which gives us um the integral from zero to H. Of route 3/4 times a over H. X squared dx. Then when we simplify this further we end up getting that. This is 3/12 3/12 A squared H. Um So that is going to be the volume in this case. And again, that's based on the side views of the pyramid, which we see her like this and like this, We see that if this is 00, the origin, then that's right. Here is point B, which is H. Over to this, right? Here is H zero. And then this right here would be H negative A over to.

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