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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.