Integration and accumulation of change practice questions (AP Calculus)

Sample questions

6 of the 7 free AP Calculus questions on this unit, from the free AP Calculus AB Practice Exam and the free AP Calculus BC Practice Exam. Each answer is behind its own toggle; the whole paper, its PDF and the full answer key are on the exam page.

1. Question 12 marks · u-substitution
   $\displaystyle\int 2x\,(x^2 + 1)^3\,dx =$
   • A.
     $(x^2 + 1)^4 + C$
   • B.
     $\frac{(x^2 + 1)^4}{4} + C$
   • C.
     $2(x^2 + 1)^4 + C$
   • D.
     $x^2 (x^2 + 1)^3 + C$
   Show answerFrom the free AP Calculus AB Practice Exam →
2. Question 22 marks · Fundamental Theorem of Calculus
   If $g(x) = \displaystyle\int_0^x \sqrt{1 + t^3}\,dt$, then $g'(2) =$
   • A.
     $3$
   • B.
     $9$
   • C.
     $\sqrt{5}$
   • D.
     $0$
   Show answerFrom the free AP Calculus AB Practice Exam →
3. Question 32 marks · Properties of definite integrals
   Given $\displaystyle\int_0^5 f(x)\,dx = 10$ and $\displaystyle\int_0^2 f(x)\,dx = 4$, find $\displaystyle\int_2^5 f(x)\,dx$.
   • A.
     $6$
   • B.
     $14$
   • C.
     $-6$
   • D.
     $40$
   Show answerFrom the free AP Calculus AB Practice Exam →
4. Question 49 marks · Accumulation (rate in / rate out)
   Accumulation. Water flows into a tank at the rate $R(t) = 20 - t^2$ liters per minute and drains out at a constant 11 liters per minute, for $0 \leq t \leq 4$. At time $t = 0$ the tank contains 30 liters.
   $R(t) = 20 - t^2, \qquad D(t) = 11, \qquad 0 \leq t \leq 4$
   (a)

   How much water flows INTO the tank during the 4 minutes?

   [2]
   (b)

   How much water is in the tank at time $t = 4$?

   [3]
   (c)

   At time $t = 2$, is the amount of water in the tank increasing or decreasing? Justify.

   [2]
   (d)

   At what time $t$ in $[0, 4]$ is the amount of water in the tank greatest? Justify, and find that maximum amount.

   [2]
   Show answerFrom the free AP Calculus AB Practice Exam →
5. Question 52 marks · Integration techniques
   $\displaystyle\int x e^{3x}\,dx =$
   • A.
     $\dfrac{e^{3x}}{9}(3x - 1) + C$
   • B.
     $\dfrac{e^{3x}}{9}(3x + 1) + C$
   • C.
     $\dfrac{x e^{3x}}{3} + C$
   • D.
     $\dfrac{x^2 e^{3x}}{6} + C$
   Show answerFrom the free AP Calculus BC Practice Exam →
6. Question 62 marks · Integration techniques
   $\displaystyle\int \frac{3x + 1}{x^2 - 1}\,dx =$
   • A.
     $2\ln|x - 1| + \ln|x + 1| + C$
   • B.
     $\ln|x - 1| + 2\ln|x + 1| + C$
   • C.
     $\tfrac{3}{2}\ln|x^2 - 1| + C$
   • D.
     $\tfrac{3}{2}\ln|x^2 - 1| + \arctan x + C$
   Show answerFrom the free AP Calculus BC Practice Exam →

The other 1 are in the free exams: the free AP Calculus AB Practice Exam and the free AP Calculus BC Practice Exam.

Key terms

The definitions and formulas AP Calculus expects for integration and accumulation of change, typeset as they appear on the paper.

Riemann sums

Approximations of the area under a curve by rectangles with left, right or midpoint heights, or by trapezoids. For an increasing function a left sum underestimates and a right sum overestimates; the exam asks which and why.

$$\int_a^b f(x)\,dx \approx \sum_{i=1}^{n} f(x_i^*)\,\Delta x$$

Definite integral

The limit of Riemann sums as the number of rectangles grows without bound; the signed area between the curve and the $x$-axis, negative where the curve is below the axis.

$$\int_a^b f(x)\,dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i)\,\Delta x$$

Fundamental Theorem of Calculus, part 1

The derivative of an accumulation function is the integrand evaluated at the upper limit, times the derivative of that limit when it is a function of $x$.

$$\frac{d}{dx} \int_a^{x} f(t)\,dt = f(x), \qquad \frac{d}{dx} \int_a^{g(x)} f(t)\,dt = f\big(g(x)\big)\,g'(x)$$

Fundamental Theorem of Calculus, part 2

A definite integral is the net change of any antiderivative between the limits. On FRQs, write the integral expression and its value; an unevaluated correct integral usually earns most of the points.

$$\int_a^b f(x)\,dx = F(b) - F(a), \qquad F' = f$$

Properties of definite integrals

Reversing the limits changes the sign, an integral over adjacent intervals adds, and constants factor out. Table and graph questions test them directly.

$$\int_a^b f = -\int_b^a f, \qquad \int_a^c f = \int_a^b f + \int_b^c f, \qquad \int_a^a f = 0$$

u-substitution

The chain rule in reverse: set $u$ equal to an inner function whose derivative appears (up to a constant) in the integrand, and change the limits when the integral is definite.

$$\int f\big(g(x)\big)\,g'(x)\,dx = \int f(u)\,du$$

Integration by parts and partial fractions (BC)

By parts reverses the product rule, for integrands such as $x e^x$ or $\ln x$. Partial fractions split a rational function with a factorable denominator into terms that integrate to logarithms.

$$\int u\,dv = uv - \int v\,du$$

Improper integral (BC)

An integral with an infinite limit or an unbounded integrand, defined as a limit. It converges if the limit is finite; write the limit explicitly.

$$\int_a^{\infty} f(x)\,dx = \lim_{b \to \infty} \int_a^b f(x)\,dx$$

Frequently asked questions

Other AP Calculus topics

• Limits and continuity
• Differentiation: definition and basic rules
• Composite, implicit and inverse functions
• Contextual applications of differentiation
• Analytical applications of differentiation
• Differential equations
• Applications of integration
• Parametric, polar and vector functions (BC)
• Infinite sequences and series (BC)

All AP Calculus topics · AP Calculus exam guide · All free practice exams