AP Calculus AB 8.7: Cross-Section Volumes

Practice AP Calculus questions on calculating volumes of solids with square or rectangular cross sections using definite integrals.

[Maximum number: 2]

The shaded region R is bounded by the graphs of the functions f and g, where $f(x)=x^{2}-2 x$

and $g(x)=x+\sin (\pi x)$ , as shown in the figure.

Figure for Question CHA-5.B—Calculate volumes of solids with known cross sections using definite integrals question 1 — AP Calculus AB

(Note: Your calculator should be in radian mode.)

Region R is the base of a solid. For this solid, at each x the cross section perpendicular to the

x-axis is a rectangle with height x and base in region R. Find the volume of the solid. Show

the setup for your calculations.

B Region R is the base of a solid. For this solid, at each x the cross section perpendicular to the x-axis is a

rectangle with height x and base in region R. Find the volume of the solid. Show the setup for your

calculations.

$\int_{0}^{3} x(g(x)-f(x)) d x$	Form of integrand	Point 3 (P3)
=7.704930<br>The volume of the solid is 7.705 (or 7.704).	Answer	Point 4 (P4)
Scoring Notes for Part B

- P3 is earned for a definite integral with an integrand presented as a product of two nonconstant

factors, with one of the factors equal to x, g(x)-f(x), or f(x)-g(x).

- The presence or absence of the differential d x will not be considered in scoring P3 or P4.

- P4 is earned for the correct answer, with or without supporting work. A reported answer should be

accurate to three places after the decimal point, rounded or truncated. An inappropriately rounded

answer does not earn the point, unless an earlier point was not earned due to inappropriate rounding.

- Incorrect or unclear communication between the integral and the correct answer is treated as scratch

work and is not considered in scoring. For example:

○ $\int_{0}^{3} x(f(x)-g(x)) d x=-7.705$ so the volume is 7.705.

Note: This response earns P3 for the integral. It also earns P4 for the correct answer.

○ $\int_{0}^{3} x(f(x)-g(x)) d x=7.705$

Note: This response earns P3 for the integral. It also earns P4 for the correct answer. (In this

instance, incorrect linkage is not considered in scoring.)

- The exact answer is $\frac{12+27 \pi}{4 \pi}$ .

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CHA-5.B—Calculate volumes of solids with known cross sections using definite integrals question 1