Q: Which of the following is NOT a correct area formula for a cross-section shape?

$$A_{\text{triangle}} = (\text{base})(\text{height})$$

Question 1

Which of the following best describes the method of slicing for finding the volume of a solid?

Accepted Answer

Dividing a solid into thin slices, approximating each slice as a cylindrical solid, and summing their volumes as the thickness approaches zero.

Question 2

Given a solid with cross-sectional area $$A(x)$$ perpendicular to the $$x-$$axis, what is the formula for the exact volume of the solid from $$x = a to$$ $$x = b$$?

Accepted Answer

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

Question 3

A loaf of bread is sliced into $$n$$ pieces of equal thickness. If the cross-sectional area of each slice is $$A(x_k)$$ and the thickness is $$\Delta x$$, which sum best approximates the total volume?

Accepted Answer

$$\sum_{k=1}^n A(x_k)\Delta x$$

Question 4

If the cross-section of a solid perpendicular to the $$x-$$axis is a square with side length $$s(x)$$, what is the area function $$A(x)$$?

Accepted Answer

$$A(x) = (s(x))^2$

Question 5

A region bounded by $$y = \sqrt{x}$$, $$x = 0$$, and $$x = 4 is $$revolved about the $$x-$$axis. Which integral gives the volume of the resulting solid?

Accepted Answer

$$V = \pi \$$int_0$$^4 (\sqrt{x})^2$ dx$$

Question 6

When using the washer method, what does the formula $$V = \pi \int_a^b [R(x)^2$ - r(x)^2$] dx$$ represent?

Accepted Answer

The volume of a solid of revolution with a hole, where $$R(x) is $$the outer radius and $$r(x) is $$the inner radius.

Question 7

A right circular cylinder of radius $$r$$ and height $$h$$ has volume:

Accepted Answer

$$V = \pi r^2$ h$$

Question 8

A region bounded by $$y = x^2$ and y = 2x$ in the first quadrant is revolved about the y$-axis. What is the correct setup for the volume?

Accepted Answer

$$V = 2\pi \$$int_0$$^2 x(2x - x^2$) dx$$

Question 9

The region between $$y = x^2$ + 1$$ and $$y = -x + 3 is $$revolved about the $$x-$$axis. Which method should be used to find the volume?

Accepted Answer

Washer method, since the region is bounded by two curves above the $$x-$$axis.

Question 10

A sphere of radius $$a is $$generated by rotating the circle $$x^2$$ + $$y^2$$ = $$a^2$$ about the $$x-$$axis. What is the volume of the sphere?

Accepted Answer

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

Question 11

If a solid has cross-sections perpendicular to the $$y-$$axis that are equilateral triangles with base $$b(y)$$, what is the area function $$A(y)$$?

Accepted Answer

$$A(y) = \frac{\sqrt{3}}{4} (b(y))^2$

Question 12

A region bounded by $$x = y^2$ + 1$$ and $$x = 3$$ for $$0 \leq y \leq \sqrt{2} is $$revolved about $$x = 3. $$What is the outer radius $$R(y)$$ for the washer method?

Accepted Answer

$$R(y) = 3 - (y^2$ + 1)$$

Question 13

Which of the following is NOT a correct area formula for a cross-section shape?

Accepted Answer

$$A_{	ext{triangle}} = (	ext{base})(	ext{height})$$

Question 14

A solid has cross-sections perpendicular to the $$x-$$axis that are right triangles with legs of length $$l(x). $$What is the area function $$A(x)$$?

Accepted Answer

$$A(x) = \frac{1}{2} (l(x))^2$

Question 15

A region bounded by $$y = \sqrt{x}$$ and $$y = 1$$ for $$1 \leq x \leq 4 is $$revolved about $$y = 1. $$What is the correct washer method setup?

Accepted Answer

Question 16

Which of the following best describes the process for finding the volume of a solid using cross-sections?

Accepted Answer

Sketch the solid and a typical cross-section, find $$A(x)$$, determine limits of integration, and integrate $$A(x)$$ over the interval.

Question 17

A region bounded by $$x = \frac{2}{y}$$ for $$1 \leq y \leq 4 is $$revolved about the $$y-$$axis. What is the correct integral for the volume?

Accepted Answer

$$V = \pi \$$int_1$$^4 \left(\frac{2}{y}\$$right)^2$$ dy$$

Volumes Using Cross-Sections: The Method of Slicing

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