Applications of Integration: Area and Volume



This section covers:


One very useful application of Integration is finding the area and volume of “curved” figures, that we couldn’t typically get without using Calculus.   Since we already know that can use the integral to get the area between the x-axis and a function, we can also get the volume of this figure by rotating the figure around either one of the axes.

Area Between Curves

Since we know how to get the area under a curve here in the Definite Integrals section, we can also get the area between two curves by subtracting the bottom curve from the top curve everywhere where the top curve is higher than the bottom curve. The cool thing about this is it even works if one of the curves is below the x-axis, as long as the higher curve always stays above the lower curve in that interval.

Note that we may need to find out where the two curves intersect (and where they intersect the x-axis) to get the limits of integration.

Here is the formal definition of the area between two curves:

Area Between Two Curves

Let’s first do problems where we’re mainly setting up:

Integration Area Problems

Notice this next problem, where it’s much easier to find the area with respect to y, since we don’t have to divide up the graph.  When we integrate with respect to y, we will have horizontal rectangles (parallel to the x-axis) instead of vertical rectangles (perpendicular to the x-axis), since we’ll use “dy” instead of “dx”.  And if we have the functions in terms of y, we need to use Inverse Functions to get them in terms of x.

Integrating To Get Area With Respect To Y

Here are more problems where we take the area with respect to y (since they are the most difficult):
More Integrating Area With Respect To Y

Volumes of Solids by Cross Sections

Now that we know how to get areas under and between curves, we can use this method to get the volume (and we’ll see later, surface area) of a three-dimensional solid.  Think about it; every day engineers are busy at work trying to figure out how much material they’ll need for certain pieces of metal, for example, and they are using calculus to figure this stuff out!

Let’s first talk about getting the volume of solids by cross-sections of certain shapes.  When doing these problems, think of the bottom of the solid being flat on your horizontal paper, and the 3-D part of it coming up from the paper.  Cross sections might be squares, rectangles, triangles, semi-circles, or trapezoids.

We could have cross-sections perpendicular to the x-axis, or y-axis; we’ll be dealing with them perpendicular to the x-axis, which is what we’re used to when differentiating with respect to x.

So here’s a definition: for cross sections with area A(x), perpendicular to the x-axis, we have \(\text{Volume = }\int\limits_{a}^{b}{{A\left( x \right)}}\,dx\).

Here are examples of volumes of cross sections between curves (slices of the volume are shown):
Volumes By Cross Sections

Volumes of Solids: The Disk Method

It turns out that the formula for the volume of a solid includes the area of the solid, since, for example, the area of a circle is \(\pi {{r}^{2}}\) and the volume of a cylinder is \(\pi {{r}^{2}}\cdot \text{height}\).   Since we know now how to get the area of a region using integration, we can get the volume of a solid by rotating the area around a line, which results in a right cylinder, or disk. Note that the radius is the distance from the axis of revolution to the function, and the “height” of each disk, or slice is “dx”:

Disk Method Drawing

Let’s so do some problems:
Volume Disk Method Problems

Volumes of Solids: The Washer Method

We learned earlier that if we have one function that is “above” another function, to get the area, we can subtract the functions and integrate.  The washer method is similar to this concept; it is just like the disk method, but it covers solids of revolution that have “holes”, where we have an inner and outer radii.

So now we have two revolving solids and we basically subtract the area of the inner solid from the area of the outer one.  Note that for this to work, the middle function must be completely inside (or touching) the outer function for the whole interval that we are integrating.

Washer Method Drawing

Let’s try some problems:
Volume Washer Method Problems

Volumes of Solids: The Shell Method

The shell method for finding volume of a solid of revolution uses integration along an axis perpendicular to the axis of revolution instead of parallel, as we’ve seen with the disk and washer methods.  The nice thing about the shell method is that you can integrate around the y-axis and not have to take the inverse of functions.  Also, the rotational solid can have a hole in it (or not), so it’s a little more robust.  It’s not intuitive though, since it deals with an infinite number of “surface areas” of rectangles in the shapes of cylinders (shells).

Here are the equations for the shell method:

Definition Of Shell Method

Since I believe the shell method is no longer required the Calculus AP tests (at least for the AB test), I will not be providing examples and pictures of this method.  Please let me know if you want it discussed further.

Learn these rules and practice, practice, practice!


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On to Integration by Parts — you are ready!