91Ó°ÊÓ

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves about the given axis. \( y = e^x \) , \( y = e^{-x} \) , \( x = 1 \) ; about the y-axis

Evaluate the integral. \( \displaystyle \int \sqrt{x} e^{\sqrt{x}}\ dx \)

We know from Example 1 that the region \( \Re = \\{ (x, y) \mid x \ge 1, 0 \le y \le \frac{1}{x} \\} \) has infinite area. Show that by rotating \( \Re \) about the x-axis we obtain a solid with finite volume.

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves about the given axis. \( y = e^{-x} \) , \( y = 0 \) , \( x = -1 \) , \( x = 0 \) ; about \( x = 1 \)

Find the volume obtained by rotating the region bounded by the curves about the given axis. \( y = \sin x \) , \( y = \cos x \) , \( 0 \le x \le \frac{\pi}{4}\) ; about \( y = 1 \)

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves about the given axis. \( y = e^x \) , \( x = 0 \) , \( y = 3 \) ; about the x-axis

Find the volume obtained by rotating the region bounded by the curves about the given axis. \( y = \sec x \) , \( y = \cos x \) , \( 0 \le x \le \frac{\pi}{3}\) ; about \( y = -1 \)

Evaluate the integral. \( \displaystyle \int \frac{1}{\sqrt{\sqrt{x} + 1}}\ dx \)

Find the area of the region under the given curve from 1 to 2. \( y = \dfrac{1}{x^3 + x} \)

Find the area of the region under the given curve from 1 to 2. \( y = \dfrac{x^2 + 1}{3x - x^2} \)