Q: A cable hangs between two towers that are 120 meters apart, forming a curve described by $$y = 20 \\cosh(0.025x). $$What is the length of the cable between the towers?

$$L = \\int_{-60}^{60} \\sqrt{1 + [0.025 \\sinh(0.025x)]^2} \\ dx$$

Q: Find the area of the surface generated when the curve $$y = \\sqrt{x}$$, $$1 \\leq x \\leq 5$$, is revolved about the x-axis.

$$A = 2\\pi \\int_{1}^{5} \\sqrt{x} \\sqrt{1 + (\\frac{1}{2}x$$^{-1/2}$$)^2$$} \\ dx$

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

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

Q: Find the volume of the solid generated when the region bounded by $$y = \\sqrt{2x - x^2$}$$ and the x-axis, for $$0 \\leq x \\leq 2$$, is revolved about the x-axis.

$$V = \\pi \\int_{0}^{2} (\\sqrt{2x - x^2$})^2$ \\ dx$$

Q: A population of a town with a 2016 population of 90,000 grows at a rate of $$P'(t) = 0.03P(t). $$What will be the population in 2021?

$$P(5) = 90,000 \\times e$$^{0.15}$$

Q: A spring with a natural length of 0.5 m is stretched to 0.8 m. If a force of 15 N is required to stretch it to 0.6 m, how much work is required to stretch it from 0.5 m to 0.8 m?

$$W = \\int_{0.5}^{0.8} 75(x - 0.5) \\ dx$$

Q: Find the arc length of the curve $$y = \\sin^3$ x$$ for $$0 \\leq x \\leq \\pi.$$

$$L = \\int_{0}^{\\pi} \\sqrt{1 + 9 \\sin^4$ x \\cos^2$ x} \\ dx$$

Q: Evaluate the integral $$\\int \\frac{e$$^{2x}$$}{1 + e$$^{2x}$$} \\ dx.$$

$$\\frac{1}{2} \\ln|1 + e$$^{2x}$$| + C$$

Q: A tank has a base area of $$4 \\ m^2$ and a height of 6$$ \\ m$$. How much work is required to pump all the water out over the top? (Assume water density 1000$$ \\ $$kg/m^3$$, $$g = 9.8 \\ m/s^2$.)

$$W = 1000 \\times 9.8 \\times \\int_{0}^{6} 4(6-y) \\ dy$$

Q: Find the volume of the solid generated when the region bounded by $$y = x^2$ and y = x$ for 0$$ \\leq x \\leq 1$$ is revolved about the y-axis.

$$V = 2\\pi \\int_{0}^{1} x(x - x^2$) \\ dx$$

Question 1

A cable hangs between two towers that are 120 meters apart, forming a curve described by $$y = 20 \cosh(0.025x). $$What is the length of the cable between the towers?

Accepted Answer

$$L = \int_{-60}^{60} \sqrt{1 + [0.025 \sinh(0.025x)]^2} \ dx$$

Question 2

Find the area of the surface generated when the curve $$y = \sqrt{x}$$, $$1 \leq x \leq 5$$, is revolved about the x-axis.

Accepted Answer

$$A = 2\pi \int_{1}^{5} \sqrt{x} \sqrt{1 + (\frac{1}{2}x$$^{-1/2}$$)^2$$} \ dx$

Question 3

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

Accepted Answer

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

Question 4

Find the volume of the solid generated when the region bounded by $$y = \sqrt{2x - x^2$}$$ and the x-axis, for $$0 \leq x \leq 2$$, is revolved about the x-axis.

Accepted Answer

$$V = \pi \int_{0}^{2} (\sqrt{2x - x^2$})^2$ \ dx$$

Question 5

A population of a town with a 2016 population of 90,000 grows at a rate of $$P'(t) = 0.03P(t). $$What will be the population in 2021?

Accepted Answer

$$P(5) = 90,000 \times e$$^{0.15}$$

Question 6

Evaluate the integral $$\int_{0}^{1} x e$$^{x^2}$$ \ dx.$$

Accepted Answer

$$\frac{1}{2}(e^1$ - 1)$$

Question 7

A spring with a natural length of 0.5 m is stretched to 0.8 m. If a force of 15 N is required to stretch it to 0.6 m, how much work is required to stretch it from 0.5 m to 0.8 m?

Accepted Answer

$$W = \int_{0.5}^{0.8} 75(x - 0.5) \ dx$$

Question 8

Find the arc length of the curve $$y = \sin^3$ x$$ for $$0 \leq x \leq \pi.$$

Accepted Answer

$$L = \int_{0}^{\pi} \sqrt{1 + 9 \sin^4$ x \cos^2$ x} \ dx$$

Question 9

Find the surface area of the sphere with radius $$r.$$

Accepted Answer

$$A = 4 \pi r^2$

Question 10

Evaluate the integral $$\int \frac{e$$^{2x}$$}{1 + e$$^{2x}$$} \ dx.$$

Accepted Answer

$$\frac{1}{2} \ln|1 + e$$^{2x}$$| + C$$

Question 11

A tank has a base area of $$4 \ m^2$ and a height of 6$$ \ m$$. How much work is required to pump all the water out over the top? (Assume water density 1000$$ \ $$kg/m^3$$, $$g = 9.8 \ m/s^2$.)

Accepted Answer

$$W = 1000 \times 9.8 \times \int_{0}^{6} 4(6-y) \ dy$$

Question 12

Find the volume of the solid generated when the region bounded by $$y = x^2$ and y = x$ for 0$$ \leq x \leq 1$$ is revolved about the y-axis.

Accepted Answer

$$V = 2\pi \int_{0}^{1} x(x - x^2$) \ dx$$

Question 13

Evaluate the integral $$\int_{0}^{1} \frac{1}{\sqrt{1 - x^2$}} \ dx.$$

Accepted Answer

$$\arcsin(1) - \arcsin(0) = \frac{\pi}{2}$$

Question 14

Find the center of mass of a thin plate in the shape of the region bounded by $$y = x^2$ and y = x$ for 0$$ \leq x \leq 1$$, assuming uniform density.

Accepted Answer

$$x_{cm} = \frac{\int_{0}^{1} x(x - x^2$) \ dx}{\int_{0}^{1} (x - x^2$) \ dx}$$

Question 15

A spherical water tank with radius $$R is $$half full. How much work is required to pump all the water to the top? (Assume water density $$\rho$$ and gravity $$g.)$$

Accepted Answer

$$W = \int_{-R}^{0} \pi (R^2$ - y^2$) \rho g (R - y) \ dy$$

Question 16

Evaluate the integral $$\int_{0}^{1} \frac{1}{(1 + x^2$)^2$$} \ dx$.

Accepted Answer

$$\frac{1}{2} (\arctan(x) + \frac{x}{1 + x^2$}) \Big|_{0}^{1}$$

Question 17

Find the surface area generated by revolving the curve $$y = \sqrt{2x - x^2$}$$, $$0 \leq x \leq 2$$, about the x-axis.

Accepted Answer

$$A = 2\pi \int_{0}^{2} \sqrt{2x - x^2$} \sqrt{1 + (\frac{1 - x}{\sqrt{2x - x^2$}})^2$} \ dx$$

Calculus II Study Guide: Arc Length, Surface Area, Physical Applications, Integrals, Population Models, and Hyperbolic Functions

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