Section 2.2 Determining Volumes by Slicing
References.
- Calculus, Early Transcendentals by Stewart, Section 6.2.
Approximating volume.
To compute the volume of a solid, we can largely mimic the strategy used for areas, starting by slicing the solid perpendicular to the rotation axis (\(x\)-axis) into thin pieces whose volume can be easily approximated. It is often natural to get the volume of a piece being “area times thickness”, with the thickness being \(\Delta x\text{.}\) If the position of each slice depends on its position \(x\) along the axis, and the area depends on the position through a function \(A(x)\text{,}\) then the slice volume is roughly \(A(x) \Delta x\text{.}\) More carefully, the approximate volume of the \(i\)-th slice is \(A(x_i^*) \Delta x\text{,}\) where \(x_i^*\) is some \(x\)-value lying within the slice.From approximate slice volume to total volume.
Once we have worked out this slice volume approximation, the rest is all much as before, with slice area \(A(x)\) here doing what slice height \(f(x)-g(x)\) did for area between curves. Leaving aside the details (seen in the text and in class), the key idea is that the total volume is given by
\begin{equation*}
V = \lim_{n \to \infty} \sum_{i=1}^n A(x_i^*) \Delta x = \int_a^b A(x) \, dx,
\end{equation*}
and we are back to the problem of finding an anti-derivative!
Finding the limits of integration: left and right extremities of the solid.
The only important ingredients missing are the limits of integration, \(a\) and \(b\text{,}\) and these are simply the positions of the left-most and right-most extremities of the solid. Sometimes these are specified as part of the description of the solid. However you often have to look at the geometry of the situation to work them out, like with computing the area between intersecting curves in the previous section.
One important detail is different for volumes: computing the area of a slice is not always as easy as computing the height of a strip. We deal with this for now by limiting ourselves to solids than can be slices into simple shapes like rectangles, circle and such. More general shapes are taken up in Calculus III, which concentrates on combining calculus with 3D geometry.
- Sketch the region under the curve \(y=\sqrt{x}\) for \(0\leq x \leq 1\text{.}\)
- Indicate rotation about the \(x\)-axis on this sketch.
- Sketch the solid obtained by rotating the above region about the \(x\)-axis.
- Sketch a typical slice used to approximate the volume; either on the above sketch or separately.
- Compute the volume of this solid.
In this very common case of a solid produced by rotation of curve \(y=f(x)\) about the \(x\)-axis, the volume formula takes the special form
\begin{equation*}
V = \int_a^b \pi r^2 dx= \pi \int_a^b [f(x)]^2 \, dx
\end{equation*}
writing \(r=f(x)\) as it is the “radius” of the solid at position \(x\text{.}\)
Example 2.2.2.
Using the strategy developed above, show that the volume of a sphere of radius \(r\) is \(\frac43\pi r^3\text{.}\)
Slices do not have to be disks, and the range of \(x\)-values is not always given explicitly; sometimes you have to find the \(x\) limits, and it helps to examine a sketch in the \(x\)-\(y\) plane.
Example 2.2.3.
- Sketch the region bounded by \(y=x\) and \(y=x^2\text{.}\)
- Indicate rotation about the \(x\)-axis on this sketch.
- Sketch the solid obtained by rotating this region about the \(x\)-axis.
- Sketch a typical slice used to approximate the volume; describe how it differs from all our previous slices!
- Compute the volume of this solid.
Slicing into washers (Annuli).
In the example above, the solid is produced by rotation of an inner curve \(y=g(x)\) and an outer curve \(y=f(x)\text{,}\) giving slices that are annuli (“washers”) with
- outer radius \(R=f(x)\text{,}\)
- inner radius \(r=g(x)\text{,}\) and
- slice area \(A = \pi(R^2-r^2)\text{.}\)
This gives volume
\begin{equation*}
V = \int_a^b \pi [R^2-r^2] dx= \pi \int_a^b \{[f(x)]^2 - [g(x)]^2\}\, dx.
\end{equation*}
Example 2.2.4. Slices do not have to be “circular” at all!
Example 2.2.5. The rotation does not have to around a coordinate axis.
- On the sketch of the plane region in Example 2.2.3, add the line \(y=-1\) and indicate rotation about this line.
- Find the volume of the solid obtained by rotating that region about the horizontal line \(y=-1\text{.}\)
Example 2.2.6. The rotation axis can be vertical instead of horizontal.
But then we want functions of \(y\) instead of \(x\text{.}\)
- Sketch the region bounded by \(y=x^2\text{,}\) \(y=8\) and \(x=0\text{.}\)
- Indicate rotation about the \(y\)-axis on this sketch.
- Sketch the solid obtained by rotating this region about the \(y\)-axis.
- Sketch a typical slice used to approximate the volume. Note that slice area depends on \(y\text{,}\) not \(x\text{.}\)
- Compute the volume of this solid.
However, in the next section we see another strategy for solids created by rotation about vertical axes which avoids equation solving.
openstax.org/books/calculus-volume-2/pages/2-2-determining-volumes-by-slicing
