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7-3 Volume as an Integral

Learning Targets

You will be able to

Concepts / Definitions

Solids of Revolution

Volume as an Integral

A solid of revolution is a solid that is formed by revolving a bounded region around a particular line. Similar to Riemann sums, the idea is to cut the solid into many pieces, then add them up.

The Disc Method

The disc method is a summation of the volume of all infinitely thin ($dx$) cylinders with radius $f(x)$ on an interval ($\int_a^b$).

\[Volume = \int_a^b\pi r^2 dx = \int_a^b\pi f(x)^2 dx\]

Note that you must account for different axes of rotation, whether it be $x = 3$ or $y = 0$, and calculate the radius (and width, as $dy$ could be used) appropriately.

Horizontal Disc Method

Vertical Disc Method

The Washer Method

The only difference between the washer and the disc method is that there is a hole to subtract. As such, the washer method is the volume of the outside function subtracted by the inside function.

\[Volume = \int_a^b\pi\ R(y)^2 dy - \int_a^b\pi\ r(y)^2 dy = \int_a^b \pi\\]

Horizontal Washer Method

Vertical Washer Method

The Shell Method

Shell Method

In some situations, we cannot calculate the volume by slicing the object into discs. In many of these cases, we can slice the object into cylindrical shells instead.

The shell method is a summation of all infinitely thin shells (think tubes) of an object.

\[Volume = \int_a^b 2\pi (r*h*w) = \int_a^b 2\pi xy\ dy\]