8.14 MC Answers and Review
6 min readβ’december 22, 2021
AP Calculus AB/BCΒ βΎοΈ
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Answers and Review for Multiple Choice Practice on Applications of Integration
βSTOP!β Before you look at the answers make sure you gave this practice quiz a try so you can assess your understanding of the concepts covered in unit 8. Click here for the practice questions: AP Calculus Unit 8 Multiple Choice Questions.
Facts about the test: Both the AP Calculus AB and BC exams have 45 multiple-choice questions and you will be given 1 hour and 45 minutes to complete the section. This means it should take you about 35 minutes to complete 15 questions.
*The following questions were not written by CollegeBoard and although they cover information outlined in the AP Calculus AB/BC Course and Exam Description, the formatting on the exam may be different.
1. Let f be the function given by f(x) = 6sin(x)cos(2x). What is the average value of f on the closed interval [5,8]?
A. -0.8237
B. -2.471
C. 0.653
D. 0.825
Answer: Use the average value formula! Don't forget to multiply the integral by 1/3
π Study AP Calculus, Unit 8.1: Finding the Average Value of a Function on an Interval
2. For time t β₯ 0, the velocity of a particle moving along the x-axis is given by v(t) = x^4 - 6x^3 + 2x^2. The initial position of the particle at time t=0 is x=6. Which of the following gives the total distance the particle travels from time t=0 to time t=5?
Answer: The answer is B. Total distance is found by integrating speed which is the absolute value of velocityΒ
π Study AP Calculus, Unit 8.2: Connecting Position, Velocity, and Acceleration of Functions Using Integrals
3.Β A magical furnace in a home consumes heating oil during a particular month at a rate modeled by the function f(t) = 0.6t^5 - 18t^3 + 16t, where f(t) is measured in gallons per day and t is the number of days since the start of the month. How many gallons of oil does the furnace consume during the first 10 days of the month?
A. 5000 gallons
B. 24267.6 gallons
C. 3.6 gallons
D. 55800 gallons
Answer: Total amount over time is the integral from 0 to 10 days. Note this function is for a magical furnace...some days it will use negative oil!
π Study AP Calculus, Unit 8.3: Using Accumulation Functions and Definite Integrals in Applied Contexts
4. Let h be the function defined by h(x) = β3x+1. Let R be the region in the first quadrant bounded above by the graph of h for 0β€xβ€4. What is the area of this?
Answer: The answer is A. Remember, the area is the integral.
π Study AP Calculus, Unit 8.4: Finding the Area Between Curves Expressed as Functions of x
5. Let p be the function defined by -8x^4 - 10x + 9 = p(x). Let R be the region in the first and second quadrant bounded above by the graph of p for -0.679β€xβ€0.704. What is the area of this?
A. 12.012
B. 13.987
C. 11.767
D. 11
Answer: The area is the integral.
π Study AP Calculus, Unit 8.6: Finding the Area Between Curves That Intersect at More Than Two Points
6. Let h be the function defined by h(x) = β3x+1. Let R be the region in the first quadrant bounded above by the graph of h for 0β€xβ€4. The region R is the base of the solid. For this solid, each cross-section perpendicular to the x-axis is a semicircle whose diameter lies in R. Find the volume of the solid.
A. (33Ο)/4
B. (33Ο)/8
C. 14Ο
D. (7Ο)/2
Answer: Half the diameter to find the radius of the semicircle then use the area formula for a circle and divide by 2.
π Study AP Calculus, Unit 8.8: Volumes with Cross Sections: Triangles and Semicircles
7. Let h be the function defined by h(x) = β3x+1. Let R be the region in the first quadrant bounded above by the graph of h for 0β€xβ€4. Let g be the antiderivative of h. Find the length of the graph of g from x=0 to x=4.
Answer: The answer is B. Arc length formula is key for this one! Don't forget to square the function.
π Study AP Calculus, Unit 8.6: Finding the Area Between Curves That Intersect at More Than Two Points
8. The base of a solid is the region in the first quadrant between the graph of y=x^5 and the x-axis for 0β€xβ€1. For the solid, each cross section perpendicular to the x-axis is a semicircle. What is the volume of the solid?
A. Ο/22
B. Ο/44
C. Ο/88
D. Ο/110
Answer: Area of a semicircle with a diameter is pi/2 times (y/2)^2 which is pi/8 y^2
π Study AP Calculus, Unit 8.8: Volumes with Cross Sections: Triangles and Semicircles
Β
9. Let R be the region in the first quadrant bounded by the graph of y=x^5, the line x=6, and the x-axis. R is the base of the solid whose cross sections perpendicular to the x-axis are equilateral triangles. What is the volume of the solid?
Answer: The answer is A. Remember the area of an equilateral triangle is sqrt3/4s^2
π Study AP Calculus, Unit 8.7: Volumes with Cross Sections: Squares and Rectangles
10. When finding the volume between curves, if there is only one function and it is revolved around the x-axis, which formula would you use?
Answer: The answer is A. Remember, when revolving around the x axis your function is in terms of y. When there is only one radius this is called the disk method.
π Study AP Calculus, Unit 8.9: Volume with Disc Method: Revolving Around the x- or y-Axis
11. When using the washer method you have ___ radii and you subtract the ___ radius from the ___ radius.
A. one; bigger; smaller
B. one; smaller; bigger
C. two; smaller; bigger
D. two; bigger; smaller
Answer: Washer method uses two radii. Think about a donut when you think about these! Subtract the outer radius from the inner one.
π Study AP Calculus, Unit 8.11: Volume with Washer Method: Revolving Around the x- or y-Axis
12. Determine the arc length of f(x) = (1/6)x^3 + (1/2)x^(-1) from x=1 to x=2:
A. 3
B. 2
C. 2.43
D. 1.43
Answer: Make sure to use the arc length formula, square root (1+(derivative)^2).
13. Determine the area of the region enclosed by y = x^2 and y=βx between x=0 and x=1.
A. 0
B. 1/3
C. -1/3
D. 2/3
Answer: Sketch an image first and you will see which functions are on top. The integral should be sqrt x-x^2 as the square root is on top.
π Study AP Calculus, Unit 8.5: Finding the Area Between Curves Expressed as Functions of y
14. Determine the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis: y=x, y=0, x=3, x=6.
A. 42.412
B. 63
C. 197.92
D. 200
Answer: Sketch this function first to visualize! Integrate after you have done a sketch of this.
π Study AP Calculus, Unit 8.6: Finding the Area Between Curves That Intersect at More Than Two Points
15. Set up the integral for the volume of the bounded region revolved about the line y=8 using the washer method. The function is y = 8/(x^2) and the bounds are y=0, x=2, and x=5.
Answer: The answer is B. When you sketch an image of this make sure to think about that it revolves around the line y=8. This line will need to be subtracted from your function as you need the space in between the two.
π Study AP Calculus, Unit 8.11: Volume with Washer Method: Revolving Around the x- or y-Axis
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