Applications of Definite Integrals Rapid Review for AP Calculus

Review the following concepts if needed:

• Average Value of a Function for AP Calculus
• Distance Traveled Problems for AP Calculus
• Definite Integral as Accumulated Change for AP Calculus
• Exponential Growth Decay Problems for AP Calculus
• Separable Differential Equations for AP Calculus
• Slope Fields for AP Calculus
• Logistic Differential Equations for AP Calculus
• Euler's Method for AP Calculus

Review Problems and Answers

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

Answer: Average value

2. Find the total distance traveled by a particle during 0 ≤ t ≤ 3 whose velocity function is shown in Figure 13.8-1.



Answer: The Total Distance Traveled

3. 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.

Answer: Total number of gallons leaked

4. How much money should Mary invest at 7.5% interest a year compounded continuously, so that she will have $100,000 after 20 years.

Answer: y (t)= y₀e^kt, k = 0.075, and t =20. y (20) =100,000 = y₀e (0.075)(20). Thus, using a calculator, you obtain y₀ ≈ 22313.

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

Answer:



6. Identify the differential equation for the slope field shown.

Answer: The slope field suggests a hyperbola of the form y² – x² = k, so and .



7. Find the solution of the initial value problem

Answer:

8. 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.

Answer: