Question 1

Which of the following integrals correctly represents the area bounded by the graphs of $$f(x) = x^2$ and g(x) = 2x$ between x = 0$ and x = 2$?

Accepted Answer

$$\$$int_0$$^2 (2x - x^2$) \, dx$$

Question 2

Given $$f(x) = \sin x$$, what is the correct integral to find the arc length of $$f(x)$$ from $$x = 0 to$$ $$x = \frac{\pi}{2}$$?

Accepted Answer

$$\$$int_0$$^{\frac{\pi}{2}} \sqrt{1 + \cos^2$ x} \, dx$$

Question 3

A region is bounded by $$y = x^2$ and y = 2x$. What is the volume of the solid generated by revolving this region about the x-axis?

Accepted Answer

$$\pi \$$int_0$$^2 \left[(2x)^2$ - (x^2$)^2$$\right] \, dx$

Question 4

Which integral represents the volume of the solid generated by revolving the region bounded by $$y = \frac{1}{x}$$ and $$y = 2x - 3$$ about the x-axis?

Accepted Answer

$$\pi \int_{a}^{b} \left[(2x - 3)^2$ - \left(\frac{1}{x}\$$right)^2$$\right] \, dx$$

Question 5

A force of 50 pounds stretches a spring 6 inches. What is the work done in stretching the spring 1 foot (12 inches) from its natural position?

Accepted Answer

$$100 ft-lb$$

Question 6

Which formula is used to calculate the arc length of a curve $$y = f(x)$$ from $$x = a to$$ $$x = b$$?

Accepted Answer

$$\int_a^b \sqrt{1 + [f'(x)]^2} \, dx$$

Question 7

If a region is bounded by $$y = x^2$ and y = 2x$, what is the correct setup for the area between the curves?

Accepted Answer

$$\$$int_0$$^2 (2x - x^2$) \, dx$$

Question 8

What is the spring constant $$k if a $$force of 50 pounds stretches a spring 6 inches?

Accepted Answer

$$k = 100 lb/ft$$

Question 9

Which of the following is the correct integral to find the work done in stretching a spring from $$x = 0 to$$ $$x = 1 ft$$, given $$k = 100 lb/ft$$?

Accepted Answer

$$\$$int_0$$^1 100x \, dx$$

Question 10

When finding the volume of a solid of revolution using the disk method, which formula is used for a region bounded by $$y = f(x)$$ from $$x = a to$$ $$x = b$$ revolved about the x-axis?

Accepted Answer

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

Question 11

If $$y = \frac{1}{x}$$ and $$y = 2x - 3$$ intersect at $$x = a$$ and $$x = b$$, which integral gives the area between the curves?

Accepted Answer

$$\int_a^b \left[(2x - 3) - \frac{1}{x}\right] \, dx$$

Question 12

Which of the following is the correct setup for the work done in stretching a spring from $$x = a to$$ $$x = b$$ with spring constant $$k$$?

Accepted Answer

$$\int_a^b kx \, dx$$

Question 13

If the region bounded by $$y = x^2$ and y = 2x$ is revolved about the y-axis, which method would you use to find the volume?

Accepted Answer

Shell method

Question 14

Which of the following represents the formula for the volume of a solid generated by revolving the region between $$y = f(x)$$ and $$y = g(x)$$ about the x-axis?

Accepted Answer

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

Question 15

A spring with spring constant $$k is $$stretched from $$x = 0 to$$ $$x = d. $$What is the work done?

Accepted Answer

$$\frac{1}{2} k d^2$

Applications of Integration: Study Notes and Practice Problems

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