Q: A population of a town with a 2016 population of $$90,000$$ grows at a rate of $$2\%$$ per year. What will be the population in 2031?

$$90,000 \times (1.02$$)^{15}$$

Question 1

Find the arc length of the curve $$y = 	an$$^{-1}$$(x)$$ from $$x = 0 to$$ $$x = 1.$$

Accepted Answer

$$\$$int_0$$^1 \sqrt{1 + \left(\frac{1}{1 + x^2$}\$$right)^2$$} \, dx$$

Question 2

A sphere of radius $$r is $$sliced by two horizontal planes at $$z = a$$ and $$z = b$$ ($$a \u003c b). $$What is the surface area of the spherical zone between these planes?

Accepted Answer

$$2\pi r (b - a)$$

Question 3

Find the area of the surface generated when the curve $$y = \sqrt{x}$$, $$0 \leq x \leq 4$$, is revolved about the $$x-$$axis.

Accepted Answer

$$2\pi \$$int_0$$^4 \sqrt{x} \sqrt{1 + \left(\frac{1}{2\sqrt{x}}\$$right)^2$$} \, dx$$

Question 4

A tank in the shape of a cone has a height of $$10 ft $$and a base radius of $$4 ft. $$How much work is required to pump all the water to the top of the tank when it is full? (Assume water weighs $$62.4 lb/ft$^3.)$$

Accepted Answer

$$62.4 \pi \$$int_0$$^{10} \left(\frac{2}{5}y\$$right)^2$$ (10 - y) \, dy$$

Question 5

Evaluate the integral $$\$$int_0$$^1 \frac{x}{\sqrt{1 - x^2$}} \, dx.$$

Accepted Answer

$$\frac{1}{2}$$

Question 6

A population of a town with a 2016 population of $$90,000$$ grows at a rate of $$2\%$$ per year. What will be the population in 2031?

Accepted Answer

$$90,000 	imes (1.02$$)^{15}$$

Question 7

Find the work required to stretch a spring from its natural length of $$10 in to a $$length of $$15 in if a $$force of $$5 lb is $$required to keep it stretched $$1 in $$beyond its natural length.

Accepted Answer

$$\int_{0}^{5} 5 \, dx$$

Question 8

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 x$-axis.

Accepted Answer

$$\pi \$$int_0$$^1 \left[(x)^2$ - (x^2$)^2$$\right] \, dx$

Question 9

Evaluate the integral $$\$$int_0$$^1 x e$$^{x^2}$$ \, dx.$$

Accepted Answer

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

Question 10

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

$$\left(\frac{3}{5}, \frac{1}{2}\right)$$

Question 11

A tank has the shape of a truncated cone with a top radius of $$3 ft$$, bottom radius of $$1 ft$$, and height $$4 ft. $$How much work is required to pump all the water out over the top when full? (Water weighs $$62.4 lb/ft$^3.)$$

Accepted Answer

$$62.4 \pi \$$int_0$$^4 \left(1 + \frac{1}{2}y\$$right)^2$$ (4 - y) \, dy$$

Question 12

Find the area of the surface generated when the curve $$y = x^3$, 0$$ \leq x \leq 1$$, is revolved about the y$-axis.

Accepted Answer

$$2\pi \$$int_0$$^1 x \sqrt{1 + (3x^2$)^2$$} \, dx$

Question 13

Evaluate the integral $$\$$int_0$$^1 \frac{dx}{\sqrt{1 - x^2$}}.$$

Accepted Answer

$$\frac{\pi}{2}$$

Question 14

A population grows according to $$P(t) = 5000e$$^{0.03t}$$, where t$ is in years. What is the population after 10$ years?

Accepted Answer

$$5000e^{0.3}$$

Question 15

Find the volume of the solid generated by revolving the region bounded by $$y = \sqrt{x}$$, $$x = 0$$, $$x = 4$$, and $$y = 0$$ about the $$x-$$axis.

Accepted Answer

$$\pi \$$int_0$$^4 x \, dx$$

Question 16

Evaluate the integral $$\$$int_0$$^1 \frac{1}{1 + x^2$} \, dx.$$

Accepted Answer

$$\frac{\pi}{4}$$

Question 17

Find the arc length of the curve $$y = x^2$ from x = 0$ to x = 1$.

Accepted Answer

$$\$$int_0$$^1 \sqrt{1 + (2x)^2$} \, dx$$

Question 18

A tank has the shape of a right circular cone with a height of $$8 ft $$and a base radius of $$3 ft. If $$the tank is filled with water, how much work is required to pump all the water to the top? (Water weighs $$62.4 lb/ft$^3.)$$

Accepted Answer

$$62.4 \pi \$$int_0$$^8 \left(\frac{3}{8}y\$$right)^2$$ (8 - y) \, dy$$

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

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