Need extra practice for your AP Calculus Unit 8 review? Outlined are the topics and applications of integration practice problems aligned with College Board’s curriculum to study for a Unit 8 test.
Unit 8 is arguably the most important unit for students in AP Calculus AB. That’s because common free response questions (FRQ) involve particle motion (8.2), accumulation functions (8.3), and area/volume (8.4-8.12). For BC students, they can expect to see an accumulation functions FRQ, but their exam rarely has an area/volume FRQ. The concepts will pop up in the multiple choice section though. Particle motion is also common for BC, but it’s typically a particle motion question given parametric equations (Unit 9) instead of functions (Unit 8).
As College Board outlines, the topics to review for Unit 8 are:
· Average Value of a Function over an Interval
Recall average rate of change of a function is the change in f(x) over the change in x. Average value is the average y-value of a function is over an interval. Essentially, you’ll find the area under f(x) divided by the change in x. Because these ideas get mixed up for students, I wanted to include both formulas below.
Often, the question is straightforward such as “Find the average value of y=sin(x) from x=0 to x=π.” If you want to practice that (please do!), your answer should be 2/π. But for a more challenging question, look at part (b) of the 2022 free response question below:
I highly recommend trying this entire problem as it is an extra practice with accumulated rates functions in 8.3.
· Position, Velocity, and Acceleration using Integrals
Everything you need to know about particle motion is listed below. This unit focuses on integrals, which means finding displacement, distance traveled, and position. To find the new position, a common mistake is forgetting to consider the given position (look back at the problem stem if it’s a free response question). Even though displacement, distance traveled, and position are the new concepts for this unit, the free response questions account for what you learned in the derivative unit as well. The review problem I chose has both derivative and integral applications mixed in.
On the AP Calculus exam, particle motion is often a calculator active problem. There are some years where it is in the no calculator section (2022). Sometimes the particle motion problem is given a table stem instead of function (2019), so try those FRQs too!
· Using Accumulation Functions and Definite Integrals in Applied Contexts
This is the most important section of this entire unit! Every year on the AP Calculus exam (AB and BC), there is a calculator active free response question covering accumulation functions. Therefore, you can get extra practice looking at past FRQs and checking your answers in the scoring guidelines.
A few key takeaways from this section: the undo a rate, you’ll take an integral. Meaning if the function provided gives liters/min and the question asks how many liters, doing an integral is essentially doing liters/min*min=liters. If the question asks if the rate is increasing or decreasing, you need to take the derivative of that function. Make sure you can evaluate both derivatives and integrals on your graphing calculator.
Finally, I like to write a new function A(t) for the amount of liters/bananas/cars at any given time. Some FRQs ask for this expression and others just ask for the greatest number (absolute maximum) of liters/bananas/cars over an interval. The general setup is below.
For some questions, the initial value is 0, or there is only one rate function. If that’s the case, you can plug in 0 or just omit that part.
Typically, the rates are given as functions of time, f(t) and g(t) for example. Notice I used a “dummy variable,” x. It’s because if I want to find the amount at t=1, I only want to plug 1 in as the upper bound of the integral, not f(1)-g(1). I want the functions to still have a variable. So I want the integral from 0 to 1 of f(x)-g(x).
3. From the 2019 AP Calculus exam:
· Area Between Curves
Your teacher may have taught area between curves with respect to x and with respect to y separately. I’m combining them here because I want you to be able to determine which is appropriate for different problems. If you see a top and bottom curve, integrate with respect to x. If you see a right and left curve, integrate with respect to y.
Try Part (a) in the problems below for practice with area. The subsequent parts are later sections of this unit so go ahead and solve those too!
4.
· Area Between Curves (Multiple Intersections)
This isn’t a very common question, finding the area between curves with multiple intersections, but if you do encounter it, you’ll have to write multiple integrals if the top and bottom (or right and left!) functions switch. See the calculator active example below.
6.
· Volume of Known Cross Sections: Squares and Rectangles
Volume is the integral of the area of the cross section. For square cross sections, the area is (side)^2. For rectangles, you’ll be given something like “each cross section is a rectangle whose height is 3 times the length of its base.” For rectangles, the area is base*height, or in this case b*3b=3b^2.
If cross sections are perpendicular to the x-axis, integrate with respect to x. If cross sections are perpendicular to the y-axis, integrate with respect to y. Just like with area, remember that integrating with respect to x will always look at Top – Bottom. Integrating with respect to y with look at Right – Left. This will give us positive distances for the sides/bases.
Try 4(b) above.
· Volume of Known Cross Sections: Triangles and Semicircles
A quick Google search for “Volume of Known Cross Sections” will give you formulas to find volume of each of the known cross sections. You can memorize the formulas, or recreate the process of drawing the region and a cross section each time. Recall the area of a triangle is 1/2bh and the area of a semicircle is 1/2π*r^2.
A volume question with cross sections of semicircles is below in part c:
7.
· Volume by Revolution: Disk Method
To determine disk vs washer method, I recommend drawing a picture of the region first, and lightly shading the region. Then draw a line from the axis of revolution to the curve(s). This represents the radius/radii of the circular cross sections.
Next, I go through a series of questions: dx or dy? If the radius is vertical, use dx. If the radius is horizontal, use dy. Is there a hole? If the line goes through some white (unshaded) space, that represents a hole, meaning that you need to do the washer method. If the region is completely up against the axis of revolution, the result of revolving would be a solid figure, so you can use disk method.
For videos and more examples of disk and washer method, or volume of known cross sections, see Flipped Math.
Typically, this unit is calculator active, but #8 can be solved without a calculator if you’re interested!
· Volume by Revolution: Washer Method
Try 5(c), 7(b), and 8(c) above.
Some teachers also go over volume using cylinders, the shell method. This is not assessed on the AP exam though. If you choose to use this method instead of disk/washer, that's totally fine, but the scoring guidelines will always use disk/washer. Plug your integral in the calculator to compare answers.
· Arc Length of a Function (BC only)
Finding arc length of a differentiable (continuous and smooth) curve f(x) from x=a to x=b requires the following formula:
If you see a question asking to find perimeter, that is arc length!
Arc length is revisited in Unit 9, finding arc length of parametric equations too.
9. Set up an integral to find the length of the curve from x=1 to x=6 for f(x)=ln(2x).
AP Calculus Unit 8 Review
For your AP Calculus Unit 8 review, make sure you are practicing integral application problems from multiple perspectives: given functions, tables, graphs, and word problems. If you need extra practice, check out the 7 best resources to study for AP Calculus tests. These resources have questions that match the rigor of your tests.
If you need further explanation on how to approach some of these difficult Unit 8 review questions, especially when all of the topics are mixed together, consider individual tutoring with me. I answer any questions students have, then provide practice solving my past test questions and previous AP exam questions. Getting more practice with problems given graphs, tables, and word problems will help you be prepared for in class tests and the AP exam.
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