# Recitation 12

Example 1: Find the exact length of curve $y=1+6x^{3/2}$, $0\leq x\leq 1$.

Hint: Use $\int_{x_1}^{x_2}\sqrt{1+(y')^2}dx$.

Problem 2: Find the exact length of curve $y=\ln(\cos x)$, $0\leq x\leq \pi/2$.

Hint: Use $\int_{x_1}^{x_2}\sqrt{1+(y')^2}dx$.

Example 3: Find the arc length function for the curve $y=2x^{3/2}$ with starting point $P_0(1,2)$.

Hint: Use $\int_{x_1}^{x_2}\sqrt{1+(y')^2}dx$.

Example 4: Find the exact area of surface obtained by rotating the curve $y=x^3, 0\leq x\leq 2$ about the $x$-axis.

Hint: Use $\int_{x_1}^{x_2}2\pi y\sqrt{1+(y')^2}dx$.

Problem 5: Find the exact area of surface obtained by rotating the curve $x=\frac{1}{3}\left(y^2+3\right)^{3/2}$ about the $x$-axis.

Hint: Use $\int_{y_1}^{y_2}2\pi y\sqrt{1+(x')^2}dy$.

Example 6: The curve $x=\sqrt{a^2-y^2}, 0\leq y\leq a/2$ about the $y$-axis. Find the area of the resulting surface.

Hint: Use $\int_{y_1}^{y_2}2\pi x\sqrt{1+(x')^2}dy$.

Example 7: If the region $R=\left\{(x,y):x\geq 1, 0\leq y\leq 1/x\right\}$ is rotated about the $x$-axis, the volume of the resulting solid is finite. Show that the surface area is infinite.

Hint: Note that $\int_1^\infty 2\pi\frac{1}{x}\sqrt{1+(y')^2}dx \geq \int_1^\infty\frac{2\pi}{x}dx=\infty$.