Question 1

A solid has cross-sectional area $$A(x)$$ perpendicular to the x-axis, and extends from $$x = a to$$ $$x = b. $$Which integral gives the volume of the solid?

Accepted Answer

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

Question 2

A pyramid 3 meters high has a square base 3 meters on a side. The cross-section perpendicular to the altitude $$x$$ meters down from the vertex is a square $$x$$ meters on a side. What is the volume of the pyramid?

Accepted Answer

Question 3

A sphere of radius $$r is $$formed by rotating the circle $$x^2$$ + $$y^2$$ = $$r^2$$ about the x-axis. Which integral gives its volume?

Accepted Answer

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

Question 4

The washer method is used to find the volume of a solid formed by rotating the region between $$y = x$$ and $$y = x^2$ + 1$$ about the x-axis from $$x = 0 to$$ $$x = 4. $$Which integral represents the volume?

Accepted Answer

Question 5

Which formula gives the volume of a solid generated by rotating $$y = f(x)$$ about the y-axis using cylindrical shells from $$x = a to$$ $$x = b$$?

Accepted Answer

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

Question 6

A tank in the shape of a cone (formed by rotating $$y = 2x$$ about the y-axis) is filled to within 2 feet of the top with olive oil weighing 57 lb/ft$$^3. $$The height of the tank is 10 feet, and the depth of the oil is 8 feet. What integral represents the work required to pump the oil to the rim?

Accepted Answer

$$W = \$$int_0$$^8 57\pi 4y^2$ (10 - y) \, dy$$

Question 7

Which formula gives the arc length $$L of a $$curve $$y = f(x)$$ from $$x = a to$$ $$x = b$$?

Accepted Answer

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

Question 8

Find the length of the curve $$y = \frac{4}{3} x$$^{3/2}$$ - 1 on $$the interval $$[0, 1]$$ using the arc length formula.

Accepted Answer

$$L = \$$int_0$$^1 \sqrt{1 + 8x$$^{1/2}$$} \, dx$$

Question 9

The lateral surface area $$S of a $$solid generated by revolving $$y = f(x)$$ about the x-axis from $$x = a to$$ $$x = b is $$given by:

Accepted Answer

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

Question 10

Gabriel’s Horn is formed by rotating $$y = 1/x$$ about the y-axis for $$y \u003c -1. $$What is the volume of the resulting solid?

Accepted Answer

$$V = \pi$$

Question 11

A solid is formed by rotating $$y = x$$ about the y-axis for $$x$$ from $$0 to$$ $$4. $$Using cylindrical shells, what is the volume?

Accepted Answer

$$V = 8\pi$$

Question 12

Which formula gives the work $$W$$ done by a variable force $$F(x)$$ moving an object from $$x = a to$$ $$x = b$$?

Accepted Answer

$$W = \int_a^b F(x) \, dx$$

Question 13

A cone is formed by rotating the line $$y = mx$$ about the x-axis from $$x = 0 to$$ $$x = h. $$What is the formula for its lateral surface area?

Accepted Answer

$$S = \pi r \sqrt{h^2$ + r^2$}$$

Question 14

A curved wedge is cut from a circular cylinder of radius 3 by two planes, one perpendicular to the axis and the other crossing at 45°. The cross-sectional area is $$A(x) = 9 - x^2$. What is the volume of the wedge from x = 0$ to x = 3$?

Accepted Answer

$$V = \$$int_0$$^3 (9 - x^2$) \, dx$$

Question 15

When using the washer method, what is the formula for the volume of a solid formed by rotating the region between two curves $$y = R(x)$$ and $$y = r(x)$$ about the x-axis from $$x = a to$$ $$x = b$$?

Accepted Answer

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

Question 16

A solid is generated by rotating $$y = 3x - x^2$ about the line x = -1$. After shifting, the equation becomes y$$ = $$-x^2 + 5x - 4$. Using cylindrical shells, what is the volume for x$ from 1$ to 4$?

Accepted Answer

$$V = 2\pi \$$int_1$$^4 (-x^3$ + 5x^2$ - 4x) \, dx$$

Question 17

Which formula gives the surface area of Gabriel’s Horn formed by revolving $$y = 1/x$$ about the x-axis for $$x in$$ $$[1, \infty)$$?

Accepted Answer

$$S = 2\pi \$$int_1$$^{\infty} \frac{1}{x} \sqrt{1 + \left(-\frac{1}{x^2$}\$$right)^2$$} \, dx$$

Question 18

In a real-world scenario, a bucket is lifted by a chain from the bottom of a well. If the force required to lift the bucket varies with height, which integral would you use to compute the total work done?

Accepted Answer

$$W = \int_{a}^{b} F(y) \, dy$$

Volumes, Surface Areas, Arc Length, and Work: Applications of Definite Integrals

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