Chapter 6 | Applications of Integration
639
⎛ ⎝ ax h
⎞ ⎠
2
A ( x ) = s 2 = ⎛ ⎝ step2 ⎞ ⎠ . Then we find the volume of the pyramid by integrating from 0 to h (step 3): V = ∫ 0 h A ( x ) dx = ∫ 0 h ⎛ ⎝ ax h ⎞ ⎠ 2 dx = a 2 h 2 ∫ 0 h x 2 dx
⎤ ⎦ | 0 h
⎡ ⎣ a 2 h 2
⎛ ⎝ 1 3
x 3 ⎞ ⎠
a 2 h .
=
= 1 3
This is the formula we were looking for.
6.6
πr 2 h for the volume of a circular cone.
Use the slicing method to derive the formula V = 1 3
Solids of Revolution If a region in a plane is revolved around a line in that plane, the resulting solid is called a solid of revolution , as shown in the following figure.
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