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.

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