
We can also use the procedure of “Slice, Approximate, Integrate” to set up integrals to compute volumes.

### Accumulation of cross-sections

We have seen how to compute certain areas by using integration. The same technique used to find those areas can be applied to find volumes as well! In this section, we consider volumes whose cross-sections taken through their bases are common shapes from geometry. In fact, we can think of these cross-sections as being “slabs” that we layer either next to each other or over each other to obtain the solid in question. We begin with a motivating example.

### Volumes of solids with known cross-sections

We can summarize the previous procedure with a simple formula that respects the geometrical reasoning used to generate the volume of a solid with a known type of cross-section:

So how do we determine which formula to use? The problem will indicate an orientation for the slice. Draw the base of the solid in the $xy$-plane, and indicate a prototypical slice on your picture. The orientation of the slice will give you the variable of integration.

Suppose that slices are taken parallel to the $y$-axis. Then, the slices are the slices are verticalthe slices are horizontal and we should integrate with respect to $x$$y$ .

Suppose that slices are taken perpendicular to the $y$-axis. Then, the slices are the slices are verticalthe slices are horizontal and we should integrate with respect to $x$$y$ .

Let’s see some examples:

Sometimes, more than one integral is needed to set up a volume of a solid with known cross-sections. Drawing a picture helps to identify when this is necessary.

### Final thoughts

To summarize some recurring ideas we have seen we have seen (and will see again), always draw and label a picture. Interpret the quantities in your picture and write down the relevant geometric quantities in terms of the variable of integration.

At the risk of being repetitive, let’s summarize some recurring ideas we have seen in the past few sections and will see again.

• Always draw and label a picture. Interpret the quantities in your picture and write down anything you need in terms of the variable of integration.
• To find vertical distances, we always take $y_{top} - y_{bot}$. To find horizontal distances, we always take $x_{right}-x_{left}$.
• When we integrate with respect to $x$, we use vertical slices and when we use vertical slices, we integrate with respect to $x$. When we integrate with respect to $y$, we use horizontal slices and when we use horizontal slices, we integrate with respect to $y$.

Remember, it takes practice to learn math. Don’t just read through examples; work them out yourself as you read along. Calculus is a hard subject. Don’t get discouraged.

“The only way to learn mathematics is to do mathematics.” — Paul Halmos