Math 2565, Spring 2020

Area and Volume

Sean Fitzpatrick
University of Lethbridge

Find the area of the region below:

If we can “slice” a solid (like a loaf of bread) and describe the (cross-sectional) area of each slice, we can reconstruct the volume:

\begin{equation*} V = \int_a^b A(x)\,dx\text{.} \end{equation*}

Examples: pyramids, cones.

Obtained by “revolving” some plane region about an axis.

Revolving the above region produces this solid

If the region contains everything between a curve and the axis of rotation, we get a disc.

\begin{equation*} A(x) = \pi r(x)^2\text{,} \end{equation*}

where \(r(x)\) is the curve-to-axis distance. (Or \(A(y)\) for a vertical axis.)

If the region is between two curves, we get a washer.

\begin{equation*} A(x) = \pi r_{\text{out}}(x)^2 - \pi r_{\text{in}}(x)^2 \end{equation*}

(inner and outer radii).

Find the volume obtained by revolving the given region as indicated:

Between \(y=4-x^2\) and the \(x\) axis, about the \(x\) axis. (About \(y=-1\text{?}\) \(y=4\text{?}\))
Between \(y=x\) and \(y=x^2\text{,}\) about the \(x\) and \(y\) axes. (About \(y=1\text{?}\) \(x=1\text{?}\))
The triangle with vetices \((0,0), (1,3), (3,1)\text{,}\) about the \(x\) axis, \(y\) axis, \(x=3\text{,}\) \(y=3\text{.}\)

Take a can with no bottom or lid, and cut vertically. You get a rectangle. Area:

\begin{equation*} L\times W = \text{ Circumference } \times \text{ Height} = 2\pi rh\text{.} \end{equation*}

We can imagine dividing a solid into nested shells:

\begin{equation*} V = \int_a^b 2\pi r(x)h(x)\,dx\text{,} \end{equation*}

but note integral with respect to \(x\) is for rotation about a vertical axis.

Shells can be useful if it's hard to get \(x\) as a function of \(y\) to do washers.

Find the volume of the solid as described. Use shells unless this is unreasonable.

Bounded by \(x=0\text{,}\) \(y=0\text{,}\) and \(y=1-x^2\text{,}\) about \(y\) axis and about \(x\) axis.
Bounded by \(x=0\text{,}\) \(y=0\text{,}\) and \(y=\cos(x)\text{,}\) about \(y\) axis and about \(x\) axis.
Bounded by \(y=x^2-2x+2\) and \(y=2x-1\text{,}\) about the \(y\) axis, \(x=1\text{,}\) and \(x=-1\text{.}\) (About the \(x\) axis?)