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
- Find the average value of y = sin x on [0, π].
- Find the total distance traveled by a particle during 0 ≤ t ≤ 3 whose velocity function is shown in Figure 13.8-1.
- 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.
- How much money should Mary invest at 7.5% interest a year compounded continuously, so that she will have $100,000 after 20 years.
- Given and y(1) = 0, solve the differential equation.
- Identify the differential equation for the slope field shown.
- Find the solution of the initial value problem
- 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: Average value
Answer: The Total Distance Traveled
Answer: Total number of gallons leaked
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.
Answer: The slope field suggests a hyperbola of the form y2 – x2 = k, so and .
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