Q: Which of the following integrals represents the perimeter of the ellipse $$\frac{x^2$}{4} + \frac{y^2$}{25} = 1$$?

$$4 \int_{0}^{\pi/2} \sqrt{4\sin^2$\theta + 25\cos^2$\theta} d\theta$$

Question 1

Which of the following integrals gives the area enclosed by the curve $$y = x^2$ and the line y = x + 2$?

Accepted Answer

$$\int_{-1}^{2} ((x + 2) - x^2$) dx$$

Question 2

Suppose you want to find the volume of the solid generated by revolving the region bounded by $$y = x^2$, y = 0$, and x = 2$ about the x$-axis. Which of the following integrals correctly represents this volume?

Accepted Answer

Question 3

Which substitution is most appropriate for evaluating $$\int \frac{2x + 1}{\sqrt{x^2$ + x}} dx$$?

Accepted Answer

$$u = x^2$ + x$$

Question 4

The area of the surface generated when $$y = \sqrt{x}$$ for $$0 \leq x \leq 4 is $$revolved about the $$x-$$axis is given by which integral?

Accepted Answer

$$2\pi \int_{0}^{4} \sqrt{x} \sqrt{1 + \left(\frac{1}{2\sqrt{x}}\$$right)^2$$} dx$$

Question 5

Which of the following integrals is equal to the length of the curve $$y = x^4$ on the interval 1$$ \leq x \leq 2$$?

Accepted Answer

$$\int_{1}^{2} \sqrt{1 + 16x^6$} dx$$

Question 6

Let $$R be $$the region bounded by $$y = x^2$, y = 0$, and x = 2$. What is the volume of the solid generated by revolving R$ about the line x = -3$?

Accepted Answer

$$2\pi \int_{0}^{2} ((x + 3)^2$ - (x^2$)^2) dx$

Question 7

Which of the following is a correct trigonometric identity useful for integration?

Accepted Answer

$$\sin^2$(x) = \frac{1 - \cos(2x)}{2}$$

Question 8

Let $$F(x) be an $$anti-derivative of $$f(x) = \frac{e$$^{x^2}$$}{x}$$ for $$x \u003e 0. $$Express $$\int_{1}^{2} \frac{e$$^{x^2}$$}{x} dx in $$terms of $$F.$$

Accepted Answer

$$F(2) - F(1)$$

Question 9

Which of the following integrals represents the perimeter of the ellipse $$\frac{x^2$}{4} + \frac{y^2$}{25} = 1$$?

Accepted Answer

$$4 \int_{0}^{\pi/2} \sqrt{4\sin^2$	heta + 25\cos^2$	heta} d	heta$$

Question 10

Let $$R be $$the region bounded by $$y = \sqrt{x}$$ and $$y = x/2 in $$the first quadrant. Which of the following describes the cross-sections perpendicular to the $$x-$$axis if the solid has squares as cross-sections?

Accepted Answer

The cross-sections perpendicular to the $$x-$$axis are squares whose sides are $$\sqrt{x} - x/2$$

Question 11

Which of the following integrals gives the arc length of the graph $$f(t) = \frac{1}{2}t$$^{3/2}$$ for 0$$ \u003c t \u003c 9$$?

Accepted Answer

$$\int_{0}^{9} \sqrt{1 + \left(\frac{3}{4}t$$^{1/2}$$\$$right)^2$$} dt$$

Question 12

Let $$R be $$the region bounded by $$y = e^x$$, $$y = 0$$, $$x = 0$$, and $$x = 1. $$What is the volume of the solid generated by revolving $$R$$ about the line $$y = 2$$?

Accepted Answer

$$\pi \int_{0}^{1} [(2^2$ - (2 - e^x)^2$)] dx$$

Question 13

Which of the following integrals gives the area of the region bounded by $$y = x^2$, y = 9$$ - $$x^2$$, and $$x = 1$$?

Accepted Answer

$$\int_{-3}^{3} ((9 - x^2$) - x^2$) dx$$

Question 14

Let $$L be $$the line segment between $$(1,2)$$ and $$(4,-4). $$Find the area of the surface generated when $$L is $$revolved around the line $$x = 2.$$

Accepted Answer

$$2\pi \int_{1}^{4} |x - 2| \sqrt{1 + \left(\frac{dy}{dx}\$$right)^2$$} dx$$

Question 15

Which of the following methods can be used to find the volume of the solid given by revolving the region bounded by $$y = x^2$, x = 2$, and the line x = -2$ about the line x = 1$?

Accepted Answer

Washer method using integration with respect to $$x$$

Question 16

Let $$R be $$the region bounded by $$y = \sec(x)$$ and $$y = 	an(x)$$ for $$0 \u003c x \u003c \frac{\pi}{4}. $$Which of the following integrals gives the area of $$R$$?

Accepted Answer

$$\int_{0}^{\frac{\pi}{4}} (\sec(x) - 	an(x)) dx$$

Question 17

Which of the following integrals gives the length of the curve $$y = \int_{0}^{x} \cos(2t) dt$$ from $$x = 0 to$$ $$x = \pi/4$$?

Accepted Answer

$$\int_{0}^{\pi/4} \sqrt{1 + \sin^2$(2x)} dx$$

Calculus II Study Guide: Applications of Integration and Volumes of Revolution

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