Integration Accumulation Worksheet with Answers – AP Calculus AB (Free PDF)

Problem 1 Left and right Riemann sums ★★★

Use four subintervals of equal width to find the left and right Riemann sums for f(x) = x² + 3 on [1, 3].

Problem 2 Trapezoidal sum from a table ★★★

Water flows into a pond at rate r(t) gallons per minute. Use a trapezoidal sum with four subintervals to estimate the water gained from t = 0 to 8 minutes.

Problem 3 Antiderivative of a polynomial ★★★

Find ∫ (6x² − 4x + 7) dx.

Problem 4 Antiderivative with ln and exponential ★★★

Find ∫ (3x + e^x − sin x) dx.

Problem 5 Meaning of a definite integral ★★★

A food truck sells burritos at a rate of s(t) burritos per hour, t hours after opening, and ∫₂⁶ s(t) dt = 148. Explain what the number 148 means.

Problem 6 Evaluating with the Fundamental Theorem ★★★

Evaluate ∫₁⁴ 3√x dx.

Problem 7 Trigonometric definite integral ★★★

Evaluate ∫₀^π (2 cos x + 1) dx.

Problem 8 Properties of definite integrals ★★★

Suppose ∫₀⁵ f(x) dx = 12 and ∫₀² f(x) dx = −3. Find

1. ∫₂⁵ f(x) dx
2. ∫₅⁰ f(x) dx
3. ∫₀⁵ (4f(x) − 2) dx

Problem 9 Substitution with a power ★★★

Find ∫ 2x(x² + 5)⁴ dx.

Problem 10 Substitution with an exponential ★★★

Find ∫ x²e^x3 dx.

Problem 11 Definite integral with substitution ★★★

Evaluate ∫₀^√7 x√(x² + 9) dx.

Problem 12 Derivative of an accumulation function ★★★

Let G(x) = ∫₁^x √(t³ + 8) dt. Find G(1) and G′(2).

Problem 13 Chain rule with an integral ★★★

Let H(x) = ∫₁^x2 (3t + 1) dt. Find H′(x) and H′(2).

Problem 14 Accumulation from a graph ★★★

The graph of f is made of line segments. Let g(x) = ∫₀^x f(t) dt. Find g(2), g(4), g(6), and the x-value where g is largest.

Problem 15 Integrating after long division ★★★

Find ∫ x² + 5x + 7x + 1 dx.

Problem 16 Completing the square ★★★

Find ∫ 1x² + 6x + 13 dx.

Problem 17 Logarithm substitution ★★★

Evaluate ∫₁^e2 ln xx dx.

Problem 18 Distance from a speed table ★★★

A cyclist’s speed v(t), in miles per hour, is recorded every 2 hours. Use a midpoint sum with three subintervals of equal width to estimate the distance traveled from t = 0 to 12 hours.

Problem 19 Displacement and distance ★★★

A particle on a line has velocity v(t) = 3t² − 12t + 9 for 0 ≤ t ≤ 4. Find its displacement and the total distance traveled.

Problem 20 Analyzing an accumulation function ★★★

Let f(x) = (x − 1)(x − 3) and g(x) = ∫₀^x f(t) dt. Find the critical points of g, classify them with the second derivative, and find g at each.