Integration and Accumulation of Change: AP Calculus AB Math Worksheet – PDF Download

AP Calculus Integration Worksheet: problems 1–5

Problem 1: Power rule antiderivative

Find \(\displaystyle\int (6x^{2}-4x+5)\,dx\).

Problem 2: Fractional and negative powers

Find \(\displaystyle\int (3\sqrt{x}-\frac{2}{x^{2}})dx\).

Problem 3: Trigonometric and exponential

Find \(\displaystyle\int (4\cos x-3e^{x}+\frac{1}{x})dx\) for \(x > 0\).

Problem 4: Initial value problem

The function \(f\) satisfies \(f^{\prime}(x)=6x^{2}-2x\) and \(f(1)=4\). Find \(f(x)\).

Problem 5: Four Riemann sums

Let \(f(x)=x^{2}\) on \([0,4]\) with \(n=4\) equal subintervals. Find the left, right, midpoint and trapezoidal sums.

AP Calculus Integration Worksheet: problems 6–10

Problem 6: Sums from a table

Maya jogs and records her speed \(v(t)\) in feet per second.

\(t\) (s)	\(0\)	\(2\)	\(5\)	\(9\)	\(10\)
\(v(t)\)	\(3\)	\(5\)	\(8\)	\(6\)	\(4\)

1. Use a left Riemann sum with the four subintervals to estimate the distance she jogs from \(t=0\) to \(t=10\).
2. Use a right Riemann sum.
3. Use a trapezoidal sum.

Problem 7: Overestimate or underestimate

For \(f(x)=\frac{1}{x}\) on \([1,5]\), which is decreasing and concave up, decide whether each estimate of \(\int_{1}^{5}\frac{1}{x}\,dx\) is an overestimate or an underestimate, and then compute the left sum with \(n=4\).

1. Left sum
2. Right sum
3. Midpoint sum
4. Trapezoidal sum

Problem 8: Properties of definite integrals

Suppose \(\int_{1}^{5} f(x)\,dx=7\), \(\int_{1}^{3} f(x)\,dx=2\) and \(\int_{1}^{5} g(x)\,dx=-3\). Find each value.

1. \(\int_{3}^{5} f(x)\,dx\)
2. \(\int_{1}^{5}(2f(x)-3g(x))dx\)
3. \(\int_{5}^{1} f(x)\,dx\)
4. \(\int_{1}^{5}(f(x)+4)dx\)

Problem 9: Integrals from geometry

The graph of \(f(x)=\sqrt{9-x^{2}}\) is a semicircle of radius 3. Use geometry to find each value.

1. \(\int_{-3}^{3} f(x)\,dx\)
2. \(\int_{0}^{3} f(x)\,dx\)
3. \(\int_{-3}^{3}(f(x)+2)dx\)

Problem 10: Evaluate with the Fundamental Theorem

Evaluate \(\displaystyle\int_{1}^{3}(4x^{3}-6x)\,dx\).

AP Calculus Integration Worksheet: problems 11–15

Problem 11: Trigonometric definite integral

Evaluate \(\displaystyle\int_{0}^{\pi/2}(3\sin x+2\cos x)\,dx\).

Problem 12: Logarithmic and exponential limits

Evaluate each integral.

1. \(\displaystyle\int_{1}^{e^{2}}\frac{3}{x}\,dx\)
2. \(\displaystyle\int_{0}^{\ln 3}4e^{x}\,dx\)

Problem 13: Derivatives of accumulation functions

Find each derivative.

1. \(\dfrac{d}{dx}\displaystyle\int_{2}^{x}\sqrt{t^{3}+1}\,dt\)
2. \(\dfrac{d}{dx}\displaystyle\int_{1}^{x^{2}}\sin t\,dt\)
3. \(\dfrac{d}{dx}\displaystyle\int_{x}^{5}\ln(t^{2}+1)\,dt\)

Problem 14: Accumulation from a graph

The graph of \(f\) consists of line segments through \((0,0)\), \((2,4)\), \((4,0)\), \((6,-3)\) and \((8,0)\). Let \(g(x)=\int_{0}^{x} f(t)\,dt\).

1. Find \(g(2)\), \(g(4)\), \(g(6)\) and \(g(8)\).
2. Find \(g^{\prime}(6)\).
3. On \([0,8]\), at what \(x\) does \(g\) have its maximum value, and what is it?
4. On what interval is the graph of \(g\) concave down?

Problem 15: Substitution with a power

Find \(\displaystyle\int 6x(3x^{2}+1)^{4}\,dx\).

AP Calculus Integration Worksheet: problems 16–20

Problem 16: Substitution with cosine

Find \(\displaystyle\int x^{2}\cos(x^{3})dx\).

Problem 17: Substitution giving a logarithm

Find \(\displaystyle\int \frac{x}{x^{2}+4}\,dx\).

Problem 18: Definite substitution

Evaluate \(\displaystyle\int_{0}^{2}3x^{2}\sqrt{x^{3}+1}\,dx\).

Problem 19: Substitution with changed limits

Evaluate \(\displaystyle\int_{0}^{\pi/2}\sin x\cos^{2}x\,dx\).

Problem 20: Long division first

Find \(\displaystyle\int \frac{x^{2}+3x+5}{x+1}\,dx\).

AP Calculus Integration Worksheet: problems 21–25

Problem 21: Completing the square

Find \(\displaystyle\int \frac{1}{x^{2}-6x+25}\,dx\).

Problem 22: Water in a tank

A tank holds 40 gallons at \(t=0\) minutes. Water flows in at the rate \(r(t)=20-2t\) gallons per minute for \(0\le t\le 8\). How many gallons are in the tank at \(t=8\)?

Problem 23: Customers entering a store

Customers enter a store at the rate \(E(t)=100\sin(\frac{\pi t}{12})\) customers per hour for \(0\le t\le 12\), where \(t\) is hours after opening. Find the total number of customers who enter, to the nearest whole number.

Problem 24: Leaking water

Water leaks from a pipe at the rate \(L(t)=6e^{-0.5t}\) gallons per hour. Find the exact amount that leaks during the first 4 hours, then round to the nearest hundredth.

Problem 25: Saving for a bike

Ethan has $200 saved and deposits money at the rate \(D(t)=40+6t\) dollars per week for 10 weeks. Find his total savings after 10 weeks.

AP Calculus Integration Worksheet: problems 26–26

Problem 26: Analyzing an accumulation function

Let \(F(x)=\int_{0}^{x}(t^{2}-2t)\,dt\).

1. Find \(F^{\prime}(x)\).
2. Find a formula for \(F(x)\) and evaluate \(F(3)\).
3. Find the critical numbers of \(F\) and classify each using \(F^{\prime\prime}\).

How to use this ap calculus integration worksheet

Print this ap calculus integration worksheet as a PDF, solve every problem on paper, then compare with the answer key.

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