When students use integrals to find areas and volumes, they can make mistakes that lead to wrong answers and confusion about important ideas. Math can be complicated, and integrals are no different.

One common mistake is not clearly identifying the shape or area they are trying to measure. When figuring out the area under a curve, it can be easy if the limits for the integral are set right. But if students skip drawing a picture of the function or the area, they can make mistakes with these limits. For example, when finding the area between two curves, students sometimes forget to check where the curves intersect. This can lead them to set the wrong intervals or leave out parts of the area they need to include.

When it comes to finding volumes, especially for shapes created by rotating a curve, students can misunderstand how to set up their integrals. For example, if they are using the disk method to find the volume around the x-axis, they need to square the function and multiply by a tiny width slice, called $dx$. The formula usually looks like this:

$V = \pi \int_{a}^{b} [f(x)]^2 \, dx.$

If a student wrongly uses the washer method when the disk method should be used (or the other way around), the volume they calculate will be wrong. Also, forgetting to consider the outer and inner radii in the washer method can lead to mistakes. Drawing a diagram can help a lot and reduce errors.

Another important part is paying close attention to the function they are working with. Sometimes, students rush their calculations without really looking at how the function behaves over the interval. For example, to find the average value of a function on the interval $[a,b