AP Calc Standards: FRQ Practice 4/6

Welcome to the second half of the AP Calc Standards Practice series, where we'll be covering some Calculator-Permitted problems in the last 3 videos of this series. This problem covers the topics of using integrals to find the area of a region bound by two curves, and some volumes of solids we can generated by using the region. Despite the fact that these will be calculator problems, you should realize that these Standards are designed to help you properly setup your answers; the difficulty will be higher on the actual exam. A vast majority of points are awarded due to the setup and demonstrating that you understand the concepts leading up to the answer, so we should only be relying on the technology towards the last minute; it will still be extremely helpful to be able to represent a sketch of what is going on with the graph in order to motivate your calculations. Be very careful when you are using the calculator because a slight difference in your input can result in a completely different calculation/resulting value. I'll emphasize the importance of representing the area of cross-sectional slices in order to calculate volumes with integration. Give yourself about 12 minutes to work this out on your own before you come back and see how it was done here.

Some Things to Keep in Mind Here:
In general, but especially when using graphing technology, I highly recommend finding the x-value of intersection between curves by graphing their DIFFERENCE and seeing when that crosses zero. Sometimes the intersection points may be way up high or low and difficult to see, so it is easier to identify zeroes located on the x-axis.

Be sure that you are clear on how to represent the area of many basic geometric shapes, since it is key to be able to formulate the cross-sectional areas whenever a region is used to make a solid. Additionally, remember that we use WASHERS whenever rotating a region around a line which is NOT one of the boundaries.

Chapters
0:00 Introduction
2:45 Visualizing the Region
3:45 Finding Points of Intersection with Desmos
4:30 Setting Up an Integral to Calculate Area
5:45 Visualizing the Solid
7:00 Representing the Area of Cross-Sectional Slices: Washers
8:00 Using Desmos to Integrate
9:30 Visualizing the New Solid
10:30 Representing the Area of Cross-Sectional Sliced: Equilateral Triangles
12:15 Using Desmos to Integrate
13:45 Some Concluding Blab