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Now that you've read up on some basic Calculus techniques, it's useful to begin going over some practice free-response questions. I know what you're thinking - it seems far too early in the year to be going over practice FRQs - but it really does help to put your new skills into practice from the very beginning. I'll work through some parts to a simple problem and then link some resources where you can find some more complex ones.
From a single glance at this FRQ, it may seem a bit overwhelming, especially since you probably haven't even taken a class yet. But don't worry - I'll help you read through this problem slowly and thoroughly so you understand every bit of it.
Let's take a look at part (a). To find the area of the space between two graphs, you have to take the integral of the "top" function and subtract the integral of the "bottom" function from it. But wait! How do you determine which function is "top" and which is "bottom", you ask?
- I find that the best way to do this is simply to plug a value of x into each function. Let's just pick x=1 as a nice and easy x-value. If we plug 1 into the sine function, we end up with just sine of pi, which is 0. So, we know that the "top" function is the sine function, since that is the function whose y-value is 0 when its x-value is 1, making the x-cubed function the "bottom" function.
Now that we've determined our "top" and "bottom" functions, we can begin with the integration process. The limits of region R are clearly shown in the picture to be x=0 and x=2, so those values will be the limits of our integration. Now, since we have our "top" and "bottom" functions and the limits of the integration, we simply subtract the integral of the x-cubed function from the integral of the sine function:
Luckily, since this question is from the calculator section of the FRQs, you don't have to work this out by hand. Using your calculator, your answer will come out to be R=4.
Now, let's work through part (b). On the exam, you'll want to actually draw the horizontal line y=-2 on the graph so you can better visualize what you're trying to find. Here, I've drawn the line y=-2 on the graph and shaded the area that the problem asks you to find:
You can see that this shaded region lies completely outside the range of the sine function, so you don't need to worry about that function at all! You do, however, need to worry about the x-cubed function.
- In this scenario, the x-cubed function will still act as your "bottom" function, and the "top" function will be the line y=-2.
- To find the limits of your integration, simply set both functions equal to each other:
- You can do this either by setting up the equation and solving it by hand or, since this is the calculator section of the FRQs, graphing each equation on your calculator and finding the points at which they intersect using 2nd>trace>5. Through either method, you should end up with x=0.539 and x=1.675.
- Your last step is to set up the integration using the limits you've found:
- Since the problem states to set up the integral and NOT to solve it, you're done!
Now that you know the basic process of reading and solving an AP-style FRQ, you can practice these throughout the year as you learn new material. As you've seen here, they may seem quite daunting at first, but they really do become quite simple once you read the problem carefully and know exactly what tools you'll use to solve it.
📌Check out these other AP Calculus resources:
For past AP Exam FRQs, answers, and scoring guidelines:
For tips and tricks for the calculator section of the exam:
For calculus techniques you should know to solve FRQs:
For things you should have memorized for the AP Exam: