Applications of integration practice questions (AP Calculus)

Sample questions

All 3 free AP Calculus questions on this unit, from the free AP Calculus AB 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 · Average value
   The average value of $f(x) = 3x^2$ on the interval $[0, 2]$ is:
   • A.
     $4$
   • B.
     $8$
   • C.
     $12$
   • D.
     $6$
   Show answerFrom the free AP Calculus AB Practice Exam →
2. Question 29 marks · Particle motion
   Particle motion. A particle moves along the x-axis so that its position at time $t$ is $x(t) = t^3 - 6t^2 + 9t$ for $0 \leq t \leq 5$, where $x$ is measured in meters and $t$ in seconds.
   $x(t) = t^3 - 6t^2 + 9t, \qquad 0 \leq t \leq 5$
   (a)

   Find the velocity $v(t)$ and acceleration $a(t)$ of the particle.

   [2]
   (b)

   On what interval(s) is the particle moving to the left? Justify your answer.

   [3]
   (c)

   At time $t = 2.5$, is the speed of the particle increasing or decreasing? Justify your answer.

   [2]
   (d)

   Find the total distance traveled by the particle from $t = 0$ to $t = 2$.

   [2]
   Show answerFrom the free AP Calculus AB Practice Exam →
3. Question 39 marks · Area & volume
   Area and volume. Let $R$ be the region in the first quadrant bounded by the graphs of $y = 4x - x^2$ and $y = x$.
   $y = 4x - x^2, \qquad y = x$
   (a)

   Find the points of intersection and the area of $R$.

   [3]
   (b)

   The region $R$ is the base of a solid whose cross-sections perpendicular to the x-axis are squares. Find the volume of the solid.

   [4]
   (c)

   Write, but do not evaluate, an integral expression for the volume of the solid generated when $R$ is rotated about the x-axis.

   [2]
   Show answerFrom the free AP Calculus AB Practice Exam →

Key terms

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

Average value of a function

The height of the rectangle over $[a, b]$ with the same area as the region under the curve. Do not confuse it with the average rate of change, which is a difference quotient.

$$f_{\text{avg}} = \frac{1}{b - a}\int_a^b f(x)\,dx$$

Net change and position

Integrating a rate gives the net change of the quantity. Position at time $b$ is the initial position plus the integral of velocity; displacement can be negative.

$$s(b) = s(a) + \int_a^b v(t)\,dt$$

Total distance travelled

The integral of speed, the absolute value of velocity, which counts motion in both directions as positive. Split the integral where velocity changes sign if integrating by hand.

$$\text{distance} = \int_a^b |v(t)|\,dt$$

Accumulation: rate in, rate out

When a quantity enters at rate $R_{in}(t)$ and leaves at rate $R_{out}(t)$, the amount at time $T$ is the initial amount plus the integral of the net rate. The amount is greatest where the net rate changes from positive to negative.

$$A(T) = A(0) + \int_0^T \big(R_{in}(t) - R_{out}(t)\big)\,dt$$

Area between curves

Integrate the top function minus the bottom function between the intersection points, or right minus left with respect to $y$ when that is simpler.

$$A = \int_a^b \big(f(x) - g(x)\big)\,dx$$

Volume by discs and washers

Rotating a region about an axis gives cross-sections that are circles (discs) or rings (washers); integrate their areas. $R$ is the outer radius and $r$ the inner radius, each measured from the axis of rotation.

$$V = \pi \int_a^b \Big(R(x)^2 - r(x)^2\Big)\,dx$$

Volume by cross-sections

If each cross-section perpendicular to the $x$-axis is a known shape (square, semicircle, equilateral triangle) with side given by the region's width, the volume is the integral of the cross-sectional area.

$$V = \int_a^b A(x)\,dx$$

Arc length (BC)

The length of the curve $y = f(x)$ from $a$ to $b$. Usually set up and evaluated with a calculator on the exam.

$$L = \int_a^b \sqrt{1 + \big(f'(x)\big)^2}\,dx$$

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
• Integration and accumulation of change
• Differential equations
• Parametric, polar and vector functions (BC)
• Infinite sequences and series (BC)

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