Q: A tank in the shape of a cylinder with radius $$3 m $$and height $$10 m is $$full of water. How much work is required to pump all the water over the top? (Assume the density of water is $$1000 kg/m$^3$$ and $$g = 9.8 m/s$^2.)$$

$$1000 \times 9.8 \times \pi \times 3^2$ \int_{0}^{10} (10 - y) \, dy$$

Q: A trough is $$10 ft $$long, $$3 ft $$wide, and $$4 ft $$deep. If the trough is full of water, what is the total work required to pump all the water over the top? (Assume the density of water is $$62.4 lb/ft$^3.)$$

$$62.4 \times 10 \times 3 \int_{0}^{4} (4 - y) \, dy$$

Q: A tank has the shape of an inverted cone with height $$10 m $$and base radius $$5 m. If $$the tank is full of water, what is the work required to pump all the water to the top? (Density of water is $$1000 kg/m$^3$$, $$g = 9.8 m/s$^2.)$$

$$1000 \times 9.8 \times \pi \int_{0}^{10} \left( \frac{5}{10} y \$$right)^2$$ (10 - y) \, dy$$

Question 1

Given the curves $$y = x^2$ and y = 4$, what is the area between these two curves from x = -2$ to x = 2$?

Accepted Answer

$$\int_{-2}^{2} (4 - x^2$) \, dx$$

Question 2

Which of the following integrals correctly represents the volume of the solid formed by revolving the region bounded by $$y = x^2$ and y = 4$ about the x$-axis from x = -2$ to x = 2$?

Accepted Answer

$$\pi \int_{-2}^{2} (4^2$ - (x^2$)^2$$) \, dx$

Question 3

A tank in the shape of a cylinder with radius $$3 m $$and height $$10 m is $$full of water. How much work is required to pump all the water over the top? (Assume the density of water is $$1000 kg/m$^3$$ and $$g = 9.8 m/s$^2.)$$

Accepted Answer

$$1000 	imes 9.8 	imes \pi 	imes 3^2$ \int_{0}^{10} (10 - y) \, dy$$

Question 4

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

Accepted Answer

Integrate $$2\pi$$ times the radius times the height of a shell with respect to $$x or$$ $$y$$

Question 5

Suppose you are asked to find the work required to lift a chain of length $$L$$ and mass per unit length $$\lambda up to $$the top of a building of height $$h. $$Which integral correctly represents the work done?

Accepted Answer

$$\int_{0}^{L} \lambda g (h - x) \, dx$$

Question 6

A trough is $$10 ft $$long, $$3 ft $$wide, and $$4 ft $$deep. If the trough is full of water, what is the total work required to pump all the water over the top? (Assume the density of water is $$62.4 lb/ft$^3.)$$

Accepted Answer

$$62.4 	imes 10 	imes 3 \int_{0}^{4} (4 - y) \, dy$$

Question 7

Which of the following is the correct setup for the volume of the solid generated by revolving the region bounded by $$y = \sqrt{x}$$, $$x = 0$$, and $$y = 2$$ about the $$x-$$axis?

Accepted Answer

$$2\pi \int_{0}^{2} y (y^2$) \, dy$$

Question 8

If $$f(x) is a $$continuous function on $$[a, b]$$, which of the following expressions gives the average value of $$f(x) on$$ $$[a, b]$$?

Accepted Answer

$$\frac{1}{b-a} \int_{a}^{b} f(x) \, dx$$

Question 9

A tank has the shape of an inverted cone with height $$10 m $$and base radius $$5 m. If $$the tank is full of water, what is the work required to pump all the water to the top? (Density of water is $$1000 kg/m$^3$$, $$g = 9.8 m/s$^2.)$$

Accepted Answer

$$1000 	imes 9.8 	imes \pi \int_{0}^{10} \left( \frac{5}{10} y \$$right)^2$$ (10 - y) \, dy$$

Question 10

Which of the following is a correct application of the washer method for finding the volume of a solid of revolution?

Accepted Answer

$$\pi \int_{a}^{b} \left( R(x)^2$ - r(x)^2$ \right) \, dx$$

Question 11

A swimming pool is $$25 m $$long, $$10 m $$wide, and $$2 m $$deep. How much work is required to pump all the water to a point $$1 m $$above the top of the pool? (Density of water is $$1000 kg/m$^3$$, $$g = 9.8 m/s$^2.)$$

Accepted Answer

$$1000 	imes 9.8 	imes 25 	imes 10 \int_{0}^{2} (3 - y) \, dy$$

Question 12

Which of the following best describes the physical meaning of the definite integral $$\int_{a}^{b} F(x) \, dx in $$the context of work?

Accepted Answer

It represents the total work done by a variable force $$F(x)$$ over the interval $$[a, b]$$

Applications of Integration – Study Notes for Exam Preparation

Guida di studio - Domande di pratica

Study guide - Flashcard