Q: Find the area of one leaf of the four-petaled rose $$r = \sin(2\theta). $$Which of the following integrals gives the correct area?

$$\frac{1}{2} \int_{0}^{\frac{\pi}{4}} \sin^2$(2\theta) \, d\theta$$

Q: Suppose you are asked to find the area enclosed by the cardioid $$r = 1 - \cos \theta. $$Which of the following integrals gives the correct area?

$$\int_{0}^{2\pi} \frac{1}{2} (1 - \cos \$$theta)^2$$ \, d\theta$$

Question 1

Which of the following integrals correctly represents the area enclosed between the curves $$f(x) = x^2$ + 2$$ and $$g(x) = 2x + 5$$?

Accepted Answer

$$\int_{-1}^{3} \left[(2x + 5) - (x^2$ + 2)\right] \, dx$$

Question 2

Find the area of one leaf of the four-petaled rose $$r = \sin(2	heta). $$Which of the following integrals gives the correct area?

Accepted Answer

$$\frac{1}{2} \int_{0}^{\frac{\pi}{4}} \sin^2$(2	heta) \, d	heta$$

Question 3

A solid has a base bounded by $$x = 2 - y^2$ and x$$ = $$y^2 - 2$, with cross-sections perpendicular to the y$-axis that are squares. Which of the following expressions gives the volume of the solid?

Accepted Answer

$$\int_{-\sqrt{2}}^{\sqrt{2}} \left[(2 - y^2$) - (y^2$ - 2)\right]^2 \, dy$$

Question 4

Which of the following integrals correctly sets up the area inside the circle $$r = 4 \cos 	heta$$ and to the right of the vertical line $$r = \sec 	heta$$?

Accepted Answer

$$\int_{-\frac{\pi}{3}}^{\frac{\pi}{3}} \frac{1}{2} \left[(4 \cos \$$theta)^2$$ - (\sec \$$theta)^2$$ight] \, d	heta$$

Question 5

Given the region bounded by $$y = x^2$, y = 8$$ - $$x^2$$, and $$x = 0 in $$the first quadrant, which of the following integrals represents the volume generated by revolving this region about the $$y-$$axis?

Accepted Answer

$$2\pi \int_{0}^{2} x \left[(8 - x^2$) - x^2$\right] \, dx$$

Question 6

Find the arc length of $$f(x) = \frac{x^2$}{2} - \frac{\ln x}{4}$$ over $$[1, 3]. $$Which of the following integrals gives the correct arc length?

Accepted Answer

$$\int_{1}^{3} \sqrt{1 + \left(x - \frac{1}{4x}\$$right)^2$$} \, dx$$

Question 7

Which of the following integrals correctly sets up the volume of the solid generated by revolving the region bounded by $$y = \ln x$$, $$y = 0$$, and $$x = e$$ about the line $$x = -2$$?

Accepted Answer

$$2\pi \int_{1}^{e} (x + 2) \ln x \, dx$$

Question 8

Suppose you are asked to find the area enclosed by the cardioid $$r = 1 - \cos 	heta. $$Which of the following integrals gives the correct area?

Accepted Answer

$$\int_{0}^{2\pi} \frac{1}{2} (1 - \cos \$$theta)^2$$ \, d	heta$$

Question 9

Given the region bounded by $$x = y^2$ and x$$ = \sqrt{y}$$, which of the following integrals represents the volume generated by revolving this region about the y$-axis?

Accepted Answer

$$2\pi \int_{0}^{1} y (\sqrt{y} - y^2$) \, dy$$

Question 10

Which of the following integrals correctly sets up the arc length of $$y = 1 - e$$^{-x}$$ over $$[0, 1]$$?

Accepted Answer

$$\int_{0}^{1} \sqrt{1 + (e$$^{-x}$$)^2$$} \, dx$

Question 11

A region is bounded by $$y = x^2$ - 1$$ and $$y = 3. If $$cross-sections perpendicular to the $$y-$$axis are equilateral triangles, which of the following expressions gives the volume of the solid?

Accepted Answer

$$\int_{-2}^{2} \frac{\sqrt{3}}{4} \left[(\sqrt{y + 1} - (-\sqrt{y + 1}))\right]^2 \, dy$$

Question 12

Which of the following integrals correctly sets up the arc length of $$f(x) = x$$^{3/2}$$ over $$[0, 2]$$?

Accepted Answer

$$\int_{0}^{2} \sqrt{1 + \left(\frac{3}{2} x$$^{1/2}$$\$$right)^2$$} \, dx$$

Question 13

Suppose you are asked to find the area of the region bounded by $$y = x^2$, y$$ = $$(x-2)^2$$, and the $$x-$$axis. Which of the following integrals gives the correct area?

Accepted Answer

$$\int_{0}^{2} \left[((x-2)^2$) - (x^2$)\right] \, dx$$

Question 14

Given the region bounded by $$f(x) = \sqrt{3} \cos(x) on$$ $$[0, \frac{\pi}{4}]$$, which of the following integrals represents the volume generated by revolving this region about the $$x-$$axis?

Accepted Answer

$$\pi \int_{0}^{\frac{\pi}{4}} 3 \cos^2$(x) \, dx$$

Question 15

Which of the following integrals correctly sets up the volume of the solid generated by revolving the region bounded by $$x = y(4 - y)$$ and $$x = (y - 2)^2$ about the x$-axis?

Accepted Answer

$$\pi \int_{0}^{4} \left([y(4 - y)]^2 - [(y - 2)^2$]^2\right) \, dy$$

Question 16

Which of the following integrals correctly sets up the volume of the solid generated by revolving the region bounded by $$y = x^2$, y = 8$$ - $$x^2$$, and $$x = 0 in $$the first quadrant about the line $$x = 3$$?

Accepted Answer

$$2\pi \int_{0}^{2} (3 - x) \left[(8 - x^2$) - x^2$\right] \, dx$$

Question 17

Which of the following integrals correctly sets up the volume of the solid generated by revolving the region bounded by $$x = y^2$ and x$$ = \sqrt{y}$$ about the line x = -1$?

Accepted Answer

$$2\pi \int_{0}^{1} [(\sqrt{y} + 1) - (y^2$ + 1)] \, dy$$

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