What AP Calculus Students Should Know for their Unit 6 Review

Jan 218 min read

Need extra practice for your AP Calculus Unit 6 review? Outlined are the topics and integration practice problems aligned with College Board’s curriculum to study for a Unit 6 test.

So far in AP Calculus, we've covered Limits and Derivatives. This unit shifts to the third main idea in Calculus, Integrals. Evaluating integrals graphically (often geometrically) involves finding the area under the curve. Evaluating integrals algebraically involves taking antiderivatives, doing the derivative rules from Unit 2 and Unit 3 in reverse.

As College Board outlines, the topics to review for Unit 6 are:

Exploring Accumulation of Change

This section establishes the contextual meaning of finding the area under a curve that represents a rate of change. You'll need to review geometry formulas to be able to find the area of rectangles, triangles, trapezoids, and circles, but the key is really being able to determine what that area under the curve represents in the given word problem and identifying the correct units. To find the correct units, think about calculating the area of a rectangle: multiply the base by the height. Do the same thing with units, and you'll multiply the independent (x-value) unit by the dependent (y-value) unit. In most cases, the rate is undone. For example, given a function representing people/hour and the x-axis representing hours, multiplying people/hour by hour cancels the hour and you're left with the number of people.

The graph below shows the rate at which water fills a bathtub, in gallons per minute. How much water has been added to the bathtub after the first 5 minutes?

ap calc unit 6 review accumulation of change graph

Approximating Areas with Riemann Sums

When the areas under a curve don't create exact geometric shapes, we can estimate the area under any curve using rectangle approximations or Riemann Sums.

Make sure you look at problems with both functions and tables for this section (tables are what you most commonly see on the AP exam!). Also, do you know how to identify if a Riemann sum is an overestimate or underestimate? A picture can help determine this but for a Calculus-based justification, you'll want to discuss whether the function is increasing or decreasing. For an increasing function, a left Riemann sum will be an underestimate and a right Riemann sum will be an overestimate. For a decreasing function, a left Riemann sum will be an overestimate and a right Riemann sum will be an underestimate.

ap calc unit 6 review riemann sums function

Riemann Sums, Summation Notation, and Definite Integral Notation

A key idea in this section is that adding up an infinite amount of rectangles (Riemann sums) will give you the exact area under the curve. Evaluating a definite integral will also give you the exact area under the curve. Problems in this section ask students to convert between summation (sigma) notation and definite integrals. I wouldn't worry too much about these problems as they aren't often assessed on the AP exam. Spending more time studying the Fundamental Theorem of Calculus and knowing your antiderivatives are more crucial for your grade on this Unit 6 test and the AP Calculus exam. Therefore, I'll combine this idea with the Fundamental Theorem of Calculus for the practice problem below.

ap calc unit 6 review riemann sums definite integral

The Fundamental Theorem of Calculus: Accumulation Functions

The Fundamental Theorem of Calculus (FTC) says that derivatives and integrals are inverse operations. In other words, derivatives and integrals undo each other. So taking the derivative of an integral will cancel the integral out.

Be aware that chain rule applies! You'll have to take the derivative of the upper bound of the integral if it is a function other than "x."

ap calc unit 6 review fundamental theorem of calculus

Interpreting the Behavior of Accumulation Functions involving Area

This section combines the previous FTC section with Unit 5 topics like increasing/decreasing, maximums/minimums, concavity, and points of inflection. These are often free response questions on the AP Calculus exam (each year typically alternates between a graph of a derivative and a graph with integrals/accumulation functions). It may be helpful to start with the justifications of each part first, then see how it applies to the given problem. In other words, a function g will always be increasing when g' is positive. Now you just need to relate g' to the given graph.

6. From the 2016 AP Calculus exam,
Applying Properties of Definite Integrals

You can search the integral properties (sum/difference, constant multiple, reverse interval, or additive interval) and find worked out examples, but the rules are pretty intuitive and shouldn't require memorization. The additive interval is the most difficult, generally, but just think if you want the area from x=1 to x=5, for example, you could find the area from x=1 to x=2 and add it to the area from x=2 to x=5.

The Fundamental Theorem of Calculus: Definite Integrals

To evaluate a definite integral analytically, take the antiderivative of the integrand function. Then plug in the upper bound to the antiderivative, and subtract the result of plugging the lower bound into the antiderivative.

You can always check these answers on your graphing calculator (although you likely won't have a calculator on this test)!

ap calc unit 6 review fundamental theorem of calc definite integrals

Finding Antiderivatives and Indefinite Integrals: Basic Rules

Hopefully you recall all of the derivative rules from Units 2 and 3. These are now done in reverse to find antiderivatives. If you don't have your derivative rules memorized, I suggest making notecards to learn them now. You can use the notecards to also help memorize antiderivatives:

ap calc unit 6 review basic antiderivative notecards

ap calc unit 6 review antiderivatives basic rules

Integrating using Substitution

Substitution or u-substitution is used to integrate a composition of functions. A composition of functions is when you have functions f(x) and g(x), and the composition creates a new function h(x) = f(g(x)). This should look familiar as taking the derivative of a composition of functions uses the chain rule.

To undo the chain rule, or to take the antiderivative of a composition of functions, you want to use u-substitution. Set u to be the inside function. Based on the function h(x) above, u = g(x). Good choices for u would be the function inside parentheses, under a radical, in the denominator, or the exponent. If you have an integral with ln(x) or arctan(x), you must set u= ln(x) or u= arctan(x) because we don’t know their antiderivatives.

I'll preface the practice question below that this is a pretty complicated u-substitution problem, so I worked out the entire solution in the answers at the end, instead of just giving the numerical solution.

Integrating using Long Division and Completing the Square
There are a few algebraic methods to rewrite the integrand in order to recognize a basic antiderivative. Sometimes it's as easy as multiplying factors together like in problem 9. More complicated algebraic approaches involve long division and completing the square.
Long division will be used when the degree in the numerator of a rational expression is greater than or equal to the degree of the denominator.

Completing the square problems can be recognized because they often have just a number in the numerator and a quadratic expression in the denominator (sometimes the quadratic expression is under a radical). Typically the answer after completing the square and integrating is an inverse trigonometric function involving u-sub. So the denominators with a radical are usually inverse sine and the denominators without a radical are inverse tangent. If there is more than just a constant in the numerator, you probably will need to integrate using u-substitution.

11.

Integrating using Integration by Parts (BC Topic)

Integration by Parts or Tabular Integration is used to integrate a product. This can be tricky because compositions of functions often present as a product: recall the derivative of f(g(x)) is f'(g(x))*g'(x). Look for a composition first (so try u-sub first) then use integration by parts.

Integration by parts isn’t too long of a process but I love taking a shortcut if I can get the same result. So, I’ll use tabular integration whenever possible. Tabular integration works if one of the factors in the product has a derivative that eventually goes to 0. Specifically, if one of the factors is a polynomial, its derivative will go to 0 and tabular integration works well.

Sometimes you must do multiple iterations of integration by parts. In that case, be consistent with your choice of u and dv. For example, if u is a trigonometric function the first time, make sure it's also the u for the second time (it may not be the exact same function; cos(x) would change to sin(x) in the second iteration).

From the AP Calculus Course and Exam Description sample problems:

Integrating using Linear Partial Fractions (BC Topic)
If you don’t recall partial fraction decomposition from Pre-Calculus, see this article. We're taking a rational function with a factorable denominator and rewriting it as a sum or difference of rational terms with linear denominators. Because the denominators are linear terms, their antiderivatives are forms of natural log.

Some log properties from Pre-Calculus come back in simplifying answers in this section. Recall that you can condense two logarithms of the same base that are being subtracted into one logarithm dividing the arguments. Or two logarithms of the same base that are being added can be rewritten into one logarithm by multiplying the arguments.

13.
$ap calc unit 6 review integrating partial fractions$

Evaluating Improper Integrals (BC Topic)

An integral is considered improper if one of the bounds is positive or negative infinity, or if the integrand is discontinuous over the interval or at an endpoint.

If you encounter an improper integral in the free response portion of the AP exam, make sure you're using correct notation, writing the limit as "a" (or any letter) approaches either infinity or the discontinuity. See my worked out solution of the problem below at the end of the post. All of the integration techniques listed come into play with improper integrals so this is a great review section to see if you can identify when to use u-sub, parts, partial fractions, or algebraic rewriting.

ap calc unit 6 review improper integrals

Selecting Techniques for Antidifferentiation

You probably noticed how similar the integrands in problems 11 and 13 are. Or maybe you wouldn't have known which technique was appropriate if this post wasn't broken down by section. If you're having difficulty determining which method of integration is correct, check out that linked post that goes into more detail with worked out examples of each.

To break down my thought process when I approach integrals, let's take a rational/fraction integrand like 13. First, I ask myself if it's a basic antiderivative that I know, like natural logarithm or an inverse trigonometric. If not, can I rewrite it so it would be inverse trig using completing the square? Is the power on top bigger than or the same as the power on bottom? Use long division. Can the denominator be factored? That's likely partial fractions. Try substitution setting u equal to the denominator.

Remember that derivatives and integrals/antiderivatives undo each other. Therefore, you can check your answer of any antiderivative by taking the derivative of your solution.

AP Calculus Unit 6 Review

For your AP Calculus Unit 6 review, make sure you are practicing integral problems from multiple perspectives: given functions, tables, and graphs. If you need extra practice, check out the 8 best resources to study for AP Calculus tests including a separate Unit 6 review for AB and BC. 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 6 review questions, especially when all of the topics are mixed together, consider individual Calculus 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.