Question 1

Which of the following integrals correctly represents the area bounded by the graphs of $$f(x) = x^2$ and g(x) = 2x$ on the interval $$[0,2]$$?

Accepted Answer

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

Question 2

Given the arc length formula $$L = \int_a^b \sqrt{1 + [f'(x)]^2} \, dx$$, which of the following represents the arc length of $$f(x) = \cos x on$$ $$[0, \frac{\pi}{2}]$$?

Accepted Answer

$$\$$int_0$$^{\frac{\pi}{2}} \sqrt{1 + \sin^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 [(2x)^2$ - (x^2$)^2$$] \, dx$

Question 4

Which of the following integrals 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, for $$x in$$ $$[1,3]$$?

Accepted Answer

$$\pi \$$int_1$$^3 \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 from its natural position?

Accepted Answer

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

Question 6

Which of the following best describes the method to find the area between two curves $$y = f(x)$$ and $$y = g(x)$$ from $$x = a to$$ $$x = b$$?

Accepted Answer

$$\int_a^b [f(x) - g(x)] \, dx$$, where $$f(x) \geq g(x) on$$ $$[a, b]$$

Question 7

If a region is revolved about the y-axis, which formula is used to find the volume using the shell method?

Accepted Answer

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

Question 8

A spring obeys Hooke's Law, $$F = kx. If a $$force of 50 pounds stretches the spring 6 inches, what is the spring constant $$k in lb/ft$$?

Accepted Answer

$$100$$

Question 9

Which of the following integrals represents the work done in stretching a spring from $$x = a to$$ $$x = b if $$the spring constant is $$k$$?

Accepted Answer

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

Question 10

Given the region bounded by $$y = x^2$ and y = 2x$, what are the x$-coordinates of the points of intersection?

Accepted Answer

$$x = 0$$ and $$x = 2$$

Question 11

Which of the following is the correct setup for the arc length of $$f(x) = \cos x$$ from $$x = 0 to$$ $$x = \frac{\pi}{2}$$?

Accepted Answer

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

Question 12

If you revolve the region bounded by $$y = x^2$ and y = 2x$ about the y-axis, which method would be most efficient?

Accepted Answer

Shell method

Applications of Integration: Study Notes and Practice Problems

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