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# Applications of Definite Integrals Rapid Review for AP Calculus

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Review the following concepts if needed:

1. Find the average value of y = sin x on [0, π].

3. Find the total distance traveled by a particle during 0 ≤ t ≤ 3 whose velocity function is shown in Figure 13.8-1.
4. Answer: The Total Distance Traveled

5. Oil is leaking from a tank at the rate of f (t) = 5e–0.1t gallons/hour, where t is measured in hours. Write an integral to find the total number of gallons of oil that will have leaked from the tank after 10 hours. Do not evaluate the integral.
6. Answer: Total number of gallons leaked

7. How much money should Mary invest at 7.5% interest a year compounded continuously, so that she will have \$100,000 after 20 years.
8. Answer: y (t)= y0ekt, k = 0.075, and t =20. y (20) =100,000 = y0e (0.075)(20). Thus, using a calculator, you obtain y0 ≈ 22313.

9. Given and y(1) = 0, solve the differential equation.

11. Identify the differential equation for the slope field shown.
12. Answer: The slope field suggests a hyperbola of the form y2x2 = k, so and .

13. Find the solution of the initial value problem

15. Use Euler's method with a step size of Δx = 0.5 to compute y (2), if y (x) is the solution of the differential equation = y + xy with initial condition y(0) = 1.

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