Calculus AB — integration and the Fundamental Theorem

The second half of AP Calculus AB turns the derivative around. This set covers ten ideas from Units 6 to 8, one question each, no calculator: a Riemann sum, accumulation, both parts of the Fundamental Theorem, substitution in a definite integral, signed area against total area, average value, an initial-value problem, a separable differential equation, and motion.

Several of these questions are built around a distinction rather than a computation, because that is where marks are actually lost. The Riemann sum question asks for the approximation AND its direction: a right sum on an increasing function over-estimates, and the monotonicity settles that before any arithmetic happens. The accumulation question asks what $\int_{0}^{5} r (t) d t$ means when $r$ is a flow rate in litres per minute — the answer is the volume ADDED over those five minutes, not the volume standing in the tank, which would need a starting amount nobody stated. Multiplying the units through (litres per minute times minutes) settles it faster than reasoning about it does. The signed-area question gives $\int_{- 1}^{1} x^{3} d x = 0$ while the area enclosed is $1/2$, and the motion question gives a displacement of zero over an interval in which the particle plainly travelled eight units. A definite integral is a signed total; an area and a distance are not.

The substitution question is the one worth slowing down for. With $u = x^{2} + 1$ the integral $\int_{0}^{2} x (x^{2} + 1)^{3} d x$ becomes $\frac{1}{2} \int_{1}^{5} u^{3} d u = 78$ — but only if the limits travel with the variable. Substituting and then evaluating between $0$ and $2$ gives $2$, a wrong answer that looks entirely reasonable on the page, which is why converting the limits at the moment of substitution is worth making a habit rather than a final check.

Every distractor is a specific, named mistake, and each explanation says which one produces it: forgetting the chain-rule factor in $\frac{d}{d x} \int_{1}^{x^{2}} \sin (t^{2}) d t$ , halving where no halving belongs, evaluating $f$ at the midpoint instead of averaging its values, integrating $4x$ as $4x^2$, dropping the constant of integration before the initial condition can pin it down. A wrong click here is meant to tell you which habit to repair.

What this does not train is the free-response half of the exam, where roughly half the marks sit and where the working carries most of the credit — setting up the integral before evaluating it, justifying a sign change, stating units. Nor does it use graphs or tables read from a figure, which real Section I questions lean on heavily. Every value here is written into the sentence instead.

Compute a left or right Riemann sum from stated values, and use monotonicity to say whether it over- or under-estimates
Read a definite integral of a rate as a net change, and check the answer by multiplying the units through
Apply the Fundamental Theorem Part 1 when the upper limit is a function of $x$, including the chain-rule factor
Evaluate a definite integral exactly with the Fundamental Theorem Part 2
Carry out a substitution in a definite integral, converting the limits along with the variable
Distinguish a signed definite integral from the total area enclosed, and displacement from total distance travelled
Compute the average value of a function on a closed interval
Solve an initial-value problem and a separable differential equation, using the initial condition to determine the constant

Built against the published Course and Exam Description structure for AP Calculus AB: Unit 6 (Integration and Accumulation of Change), Unit 7 (Differential Equations) and Unit 8 (Applications of Integration — average value, area, motion). The exam runs 3 hours 15 minutes across two sections, multiple choice and free response, each with a calculator and a no-calculator part; this set is no-calculator throughout, so every number resolves exactly. Every function, number and scenario here is invented for this material — nothing is reproduced from any College Board publication, released exam or scoring guideline. Zestly is an independent study tool. It is not affiliated with the College Board, which owns the AP Calculus AB exam, and it is not an exam centre.

Sample question

A function $f$ is strictly increasing on $[0, 4]$, with $f(0) = 1$, $f(2) = 3$ and $f(4) = 7$. Using a right Riemann sum with the two subintervals $[0, 2]$ and $[2, 4]$, what is the approximation of $\int_0^4 f(x)\,dx$, and how does it compare with the true value?

$8$, an over-estimate
$20$, an over-estimate
$20$, an under-estimate
$8$, an under-estimate

See the answer

$20$, an over-estimate

Each subinterval has width $2$, and a right sum uses the value at the right-hand endpoint: $2f(2) + 2f(4) = 2(3) + 2(7) = 6 + 14 = 20$. The direction follows from the monotonicity, not from the arithmetic: on an increasing function the right endpoint is the largest value on its subinterval, so every rectangle rises above the curve and the sum runs high. The value $8$ is the LEFT sum, $2f(0) + 2f(2) = 2 + 6$, which for the same reason runs low — a correct statement about a different sum, and an answer to a question that was not asked. Notice which data each sum uses: $f(4)$ enters the right sum and $f(0)$ never does, while the left sum is the other way round.

Try this quiz →

← Calculus AB

↑ AP