Question 1

Which formula represents the volume of a solid with a constant cross-sectional area A and height h?

Accepted Answer

$$V = A h$$

Question 2

A solid is sliced perpendicular to the x-axis, and the area of the cross-section at $$x is$$ $$A(x). $$Which integral gives the exact volume of the solid between $$x = a$$ and $$x = b$$?

Accepted Answer

$$V = \int_a^b A(x) dx$$

Question 3

The cross-sectional area of a sphere of radius $$r at $$position $$x is$$ $$A(x) = \pi (r^2$ - x^2$). $$What is the integral expression for the volume of the sphere?

Accepted Answer

$$V = \int_{-r}^r \pi (r^2$ - x^2$) dx$$

Question 4

If a solid is divided into $$n$$ slabs of width $$\Delta x$$, and the area of the cross-section at $$x_i^* is$$ $$A(x_i^*)$$, which sum approximates the volume?

Accepted Answer

$$V \approx \sum_{i=1}^n A(x_i^*) \Delta x$$

Question 5

A rectangular box has length $$l$$, width $$w$$, and height $$h. $$What is its volume?

Accepted Answer

$$V = l w h$$

Question 6

Which formula is used to find the area of a cross-section that is a disk of radius $$r$$?

Accepted Answer

$$A = \pi r^2$

Question 7

For a solid of revolution with a washer-shaped cross-section, how is the area of the washer calculated?

Accepted Answer

$$A = \pi (r_{out}^2 - r_{in}^2)$$

Question 8

A solid is formed by revolving the region between $$y = f(x)$$ and $$y = g(x)$$ about the x-axis. What is the formula for the volume using washers?

Accepted Answer

$$V = \int_a^b \pi \left( [f(x)]^2 - [g(x)]^2 \right) dx$$

Question 9

Which of the following best describes the process of finding the volume of a solid using calculus?

Accepted Answer

Dividing the solid into thin slabs, approximating each slab's volume, and taking the limit as the slabs become infinitely thin.

Question 10

If the cross-sectional area $$A(x) of a $$solid is constant for all $$x in$$ $$[a, b]$$, what does the definite integral $$\int_a^b A(x) dx$$ simplify to?

Accepted Answer

$$A (b - a)$$

Question 11

A hospital needs to calculate the volume of a medication capsule shaped like a solid of revolution. If the cross-sectional area at $$x is$$ $$A(x)$$, which formula should they use?

Accepted Answer

$$V = \int_a^b A(x) dx$$

Question 12

Which of the following is NOT a plausible method for approximating the volume of a solid using calculus?

Accepted Answer

Measuring only the perimeter of the solid.

Question 13

A solid is bounded between $$x = a$$ and $$x = b$$, and its cross-sectional area is $$A(x). $$Which statement is true?

Accepted Answer

The volume is given by $$V = \int_a^b A(x) dx.$$

Question 14

If a sphere has radius $$r$$, what is its exact volume?

Accepted Answer

$$V = \frac{4}{3} \pi r^3$

Question 15

Which of the following is a real-world application of finding volumes using definite integrals?

Accepted Answer

Calculating the volume of a water tank with a non-uniform cross-section.

Question 16

A solid of revolution is generated by revolving $$y = f(x)$$ about the x-axis. What is the formula for the volume if the cross-section is a disk?

Accepted Answer

$$V = \int_a^b \pi [f(x)]^2 dx$$

Question 17

Which step is necessary to obtain the exact volume of a solid using the method of cross-sections?

Accepted Answer

Take the limit as the number of slabs approaches infinity.

Question 18

In the context of solids of revolution, what is the difference between a disk and a washer?

Accepted Answer

A disk has no hole, while a washer has a hole in the center.

Question 19

A solid is formed by revolving the region between $$y = f(x)$$ and $$y = g(x)$$ about the y-axis. Which integral gives its volume?

Accepted Answer

$$V = \int_c^d \pi \left( [f(y)]^2 - [g(y)]^2 \right) dy$$

Applications of Definite Integrals: Volumes Using Cross-Sections

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