Question 1

Given the region $$R$$ under the curve $$y = f(x)$$ from $$x = a to$$ $$x = b (as $$shown in the first image), which of the following integrals correctly represents the area of $$R$$?

Accepted Answer

$$\int_a^b f(x)\,dx$$

Question 2

Suppose you want to find the area between the curves $$y = f(x)$$ and $$y = g(x)$$ from $$x = a to$$ $$x = b (as $$shown in the third image). Which integral gives the correct area?

Accepted Answer

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

Question 3

A solid has a cross-sectional area $$A(x)$$ perpendicular to the $$x-$$axis from $$x = a to$$ $$x = b (as $$shown in the fourth image). What is the formula for the volume of the solid?

Accepted Answer

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

Question 4

If the region between $$y = f(x)$$ and $$y = g(x)$$ from $$x = a to$$ $$x = b is $$revolved about the $$x-$$axis (as in the fifth image), which integral gives the volume of the resulting solid?

Accepted Answer

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

Question 5

Given the region bounded by $$x = g(y)$$ and $$x = f(y)$$ from $$y = c to$$ $$y = d (as in $$the second image), what is the area of this region?

Accepted Answer

$$A = \int_c^d [f(y) - g(y)] dy$$

Question 6

A region bounded by $$x = p(y)$$ and $$x = q(y) is $$revolved about the $$y-$$axis (see the eighth image). What is the formula for the volume of the solid generated?

Accepted Answer

$$V = \pi \int_c^d [p(y)^2$ - q(y)^2$] dy$$

Question 7

Which of the following integrals represents the arc length of a smooth curve $$y = f(x)$$ from $$x = a to$$ $$x = b (as $$shown in the sixteenth image)?

Accepted Answer

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

Question 8

Suppose you revolve the curve $$y = f(x)$$ from $$x = a to$$ $$x = b$$ about the $$x-$$axis (as in the seventeenth image). What is the formula for the surface area of the resulting solid?

Accepted Answer

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

Question 9

A thin wire of length $$L$$ has a density function $$\rho(x)$$ for $$a \leq x \leq b. $$What is the total mass of the wire?

Accepted Answer

$$m = \int_a^b \rho(x) dx$$

Question 10

A region bounded by $$y = f(x)$$ and $$y = g(x)$$ from $$x = a to$$ $$x = b is $$revolved about the $$y-$$axis (as in the twelfth image). Which method and formula would you use to find the volume?

Accepted Answer

Shell method: $$V = 2\pi \int_a^b x [f(x) - g(x)] dx$$

Question 11

A tank in the shape of a solid of revolution is filled with water. The cross-sectional area at height $$y is$$ $$A(y)$$, and the water must be lifted to a platform at height $$L. $$What is the work required to pump all the water to the platform?

Accepted Answer

$$W = \$$int_0$$^L \rho g A(y) (L - y) dy$$

Question 12

Which of the following integrals gives the future position $$s(t) of an $$object with initial position $$s_0$$ and velocity $$v(x)$$?

Accepted Answer

$$s(t) = s_0$ + \$$int_0$$^t v(x) dx$$

Question 13

If $$Q'(t) is $$the rate of change of a quantity $$Q(t)$$, what is the net change in $$Q$$ from $$t = a to$$ $$t = b$$?

Accepted Answer

$$Q(b) - Q(a) = \int_a^b Q'(t) dt$$

Question 14

A region bounded by $$x = p(y)$$ and $$x = q(y)$$ from $$y = c to$$ $$y = d is $$revolved about the $$y-$$axis. Which method and formula would you use to find the volume?

Accepted Answer

Washer method: $$V = \pi \int_c^d [p(y)^2$ - q(y)^2$] dy$$

Question 15

Which of the following integrals represents the surface area of the solid generated by revolving $$x = g(y)$$ from $$y = c to$$ $$y = d$$ about the $$y-$$axis?

Accepted Answer

$$S = \int_c^d 2\pi g(y) \sqrt{1 + [g'(y)]^2} dy$$

Chapter 6: Applications of Integration – Area, Volume, Arc Length, and Physical Applications

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