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### Section 8.2 : Surface Area

1. Set up, but do not evaluate, an integral for the surface area of the object obtained by rotating $$y = 7x + 2$$ , $$- 5 \le y \le 0$$ about the $$x$$-axis using,
1. $$\displaystyle ds = \sqrt {1 + {{\left[ {\frac{{dy}}{{dx}}} \right]}^2}} \,dx$$
2. $$\displaystyle ds = \sqrt {1 + {{\left[ {\frac{{dx}}{{dy}}} \right]}^2}} \,dy$$
2. Set up, but do not evaluate, an integral for the surface area of the object obtained by rotating $$y = 1 + 2{x^5}$$ , $$0 \le x \le 1$$ about the $$x$$-axis using,
1. $$\displaystyle ds = \sqrt {1 + {{\left[ {\frac{{dy}}{{dx}}} \right]}^2}} \,dx$$
2. $$\displaystyle ds = \sqrt {1 + {{\left[ {\frac{{dx}}{{dy}}} \right]}^2}} \,dy$$
3. Set up, but do not evaluate, an integral for the surface area of the object obtained by rotating $$x = {{\bf{e}}^{2y}}$$ , $$- 1 \le y \le 0$$ about the $$y$$-axis using,
1. $$\displaystyle ds = \sqrt {1 + {{\left[ {\frac{{dy}}{{dx}}} \right]}^2}} \,dx$$
2. $$\displaystyle ds = \sqrt {1 + {{\left[ {\frac{{dx}}{{dy}}} \right]}^2}} \,dy$$
4. Set up, but do not evaluate, an integral for the surface area of the object obtained by rotating $$y = \cos \left( {\displaystyle \frac{1}{2}x} \right)$$ , $$0 \le x \le \pi$$ about
1. the $$x$$-axis
2. the $$y$$-axis
5. Set up, but do not evaluate, an integral for the surface area of the object obtained by rotating $$x = \sqrt {3 + 7y}$$ , $$0 \le y \le 1$$ about
1. the $$x$$-axis
2. the $$y$$-axis
6. Find the surface area of the object obtained by rotating $$\displaystyle y = \frac{1}{4}\sqrt {6x + 2}$$ , $$\displaystyle \frac{{\sqrt 2 }}{2} \le y \le \frac{{\sqrt 5 }}{2}$$ about the $$x$$-axis.
7. Find the surface area of the object obtained by rotating $$y = 4 - x$$ , $$1 \le x \le 6$$ about the $$y$$-axis.
8. Find the surface area of the object obtained by rotating $$x = 2y + 5$$ , $$- 1 \le x \le 2$$ about the $$y$$-axis.
9. Find the surface area of the object obtained by rotating $$x = 1 - {y^2}$$ , $$0 \le y \le 3$$ about the $$x$$-axis.
10. Find the surface area of the object obtained by rotating $$x = {{\bf{e}}^{2y}}$$ , $$- 1 \le y \le 0$$ about the $$y$$-axis.
11. Find for the surface area of the object obtained by rotating $$y = \cos \left( {\displaystyle \frac{1}{2}x} \right)$$ , $$0 \le x \le \pi$$ about the $$x$$-axis.