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

AP Calculus Integration Test: questions 1–5

Question 1: Estimating from a table (3 points)

Cars enter a parking garage at the rate \(R(t)\) cars per hour, where \(t\) is the number of hours after 6 A.M. Values of \(R\) are measured at five times .

1. Use a left Riemann sum with four subintervals to approximate \(\int_{0}^{8} R(t)\,dt\). (1 point)
2. Use a trapezoidal sum with the same four subintervals to approximate \(\int_{0}^{8} R(t)\,dt\). (1 point)
3. Explain the meaning of \(\int_{0}^{8} R(t)\,dt\) in the context of the problem. Include units. (1 point)

Question 2: Properties of integrals (2 points)

The functions \(f\) and \(g\) are continuous, with \(\int_{0}^{4} f(x)\,dx=10\), \(\int_{4}^{7} f(x)\,dx=-3\) and \(\int_{0}^{7} g(x)\,dx=6\).

1. Find \(\int_{0}^{7} f(x)\,dx\). (1 point)
2. Find \(\int_{0}^{7}(3f(x)-g(x)+2)dx\). (1 point)

Question 3: Accumulation from a graph (3 points)

The graph of \(f\) consists of line segments and is shown on \([0,8]\). Let \(g(x)=\int_{0}^{x} f(t)\,dt\).

1. Find \(g(6)\). (1 point)
2. Find \(g^{\prime}(5)\). (1 point)
3. Find the value of \(x\) in \([0,8]\) at which \(g\) has its absolute maximum. Justify your answer. (1 point)

Question 4: Fundamental Theorem evaluation (2 points)

Evaluate \(\displaystyle\int_{1}^{4}(3\sqrt{x}+\frac{2}{x^{2}})dx\). Show the antiderivative you use.

Question 5: Substitution (2 points)

Find each integral.

1. \(\displaystyle\int 4x(x^{2}-5)^{3}\,dx\) (1 point)
2. \(\displaystyle\int_{1}^{e}\frac{(\ln x)^{2}}{x}\,dx\) (1 point)

AP Calculus Integration Test: questions 6–8

Question 6: Water in a tank (3 points)

A tank contains 30 gallons of water at time \(t=0\) minutes. Water enters the tank at the rate \(r(t)=12-2t\) gallons per minute for \(0\le t\le 8\) (a negative rate means water is being pumped out).

1. Write an expression for \(V(t)\), the number of gallons in the tank at time \(t\), and simplify it. (1 point)
2. At what time \(t\) is the amount of water in the tank greatest? Justify your answer, and find the maximum amount. (1 point)
3. Find \(V(8)\). Is the amount of water increasing or decreasing at \(t=8\)? Explain. (1 point)

Question 7: Division and completing the square (3 points)

Find each integral.

1. \(\displaystyle\int \frac{x^{2}-1}{x+3}\,dx\) (2 points)
2. \(\displaystyle\int \frac{1}{x^{2}+6x+10}\,dx\) (1 point)

Question 8: Differentiating an integral (2 points)

Let \(G(x)=\displaystyle\int_{1}^{x^{2}}(t+3)\,dt\).

1. Find \(G^{\prime}(x)\). (1 point)
2. Find \(G^{\prime}(2)\). (1 point)

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Take this ap calculus integration test without notes, then grade it with the test answer key out of 20 points.

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