This ap calculus integration worksheet gives 26 problems on integration and accumulation of change, ordered from easy to hard, with a separate answer key.
These 26 problems practice AP Calculus AB Unit 6: antiderivatives, Riemann sums, definite integral properties, the Fundamental Theorem of Calculus, accumulation functions, and u-substitution, long division and completing the square.
Problems move from basic antiderivatives to applied accumulation questions. Show every antiderivative and every substitution before you evaluate limits.
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) | \(0\) | \(2\) | \(5\) | \(9\) | \(10\) |
|---|---|---|---|---|---|
| \(v(t)\) | \(3\) | \(5\) | \(8\) | \(6\) | \(4\) |
- Use a left Riemann sum with the four subintervals to estimate the distance she jogs from \(t=0\) to \(t=10\).
- Use a right Riemann sum.
- 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 \(n=4\).
- Left sum
- Right sum
- Midpoint sum
- 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.
- \(\int_{3}^{5} f(x)\,dx\)
- \(\int_{1}^{5}(2f(x)-3g(x))dx\)
- \(\int_{5}^{1} f(x)\,dx\)
- \(\int_{1}^{5}(f(x)+4)dx\)
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.
- \(\int_{-3}^{3} f(x)\,dx\)
- \(\int_{0}^{3} f(x)\,dx\)
- \(\int_{-3}^{3}(f(x)+2)dx\)
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.
- \(\displaystyle\int_{1}^{e^{2}}\frac{3}{x}\,dx\)
- \(\displaystyle\int_{0}^{\ln 3}4e^{x}\,dx\)
Problem 13: Derivatives of accumulation functions
Find each derivative.
- \(\dfrac{d}{dx}\displaystyle\int_{2}^{x}\sqrt{t^{3}+1}\,dt\)
- \(\dfrac{d}{dx}\displaystyle\int_{1}^{x^{2}}\sin t\,dt\)
- \(\dfrac{d}{dx}\displaystyle\int_{x}^{5}\ln(t^{2}+1)\,dt\)
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 \(g(x)=\int_{0}^{x} f(t)\,dt\).
- Find \(g(2)\), \(g(4)\), \(g(6)\) and \(g(8)\).
- Find \(g^{\prime}(6)\).
- On \([0,8]\), at what \(x\) does \(g\) have its maximum value, and what is it?
- 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 \(F(x)=\int_{0}^{x}(t^{2}-2t)\,dt\).
- Find \(F^{\prime}(x)\).
- Find a formula for \(F(x)\) and evaluate \(F(3)\).
- 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.
See every chapter on the AP Calculus AB math page.