WEBVTT
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suppose you want to find the volume of a right
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circular cone with height H. And base radius R
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. To do this we draw on the partition plane
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R. Right triangle with high H. And base
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radius R. Now looking at this, we see
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that the triangle is made up of the line.
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Y. Equal to H. And this line whose
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points are this would be Rh And this will be
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the origin 00. And so the lion formed by
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these two points would be why equal to that's a
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rice overrun of H. Over our. And then
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times X. Now if we rotate this right triangle
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about the why access can then form the right circular
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cone. And so using a strip that is parallel
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to the axis of rotation, the solid form by
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rotating this trip about the Y axis, it's a
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cylinder with radius equal to X. And height.
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That is the difference between the top function H.
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And the bottom function H. Over our X.
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And so by cylindrical shell method we have V equal
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to two pi times the integral from 0 to our
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. Since our area is found over the X interval
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zero to our and then you have radius which is
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X times height which is H minus H over our
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X. And then dx. And then simplifying we
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have two pi Integral from 0 to our of X
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. H minus H. Over our X squared dx
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. And so integrating with respect to X we have
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two pi times We have aged times x squared over
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two minus H. Over our times X cube over
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three. This evaluated from 0 to our So we
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have two pi times When exes are we have age
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times R squared over two minus H over our times
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are cube over three and when X zero everything becomes
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zero. So from here we have two pi times
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each over two R squared minus H over three R
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squared. And this is just two pi times each
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over six times R squared. Or this is just
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by over three H times R squared. Therefore the
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volume of the right circular cone is by over three
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R squared times each.