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## Educators

Rk ss RG
+ 17 more educators

### Problem 1

Let $S$ be the solid obtained by rotating the region shown in the figure about the y-axis. Explain why it is awkward to use slicing to find the volume $V$ of $S$. Sketch a typical approximating shell. What are its circumference and height? Use shells to find $V$. Amrita B.

### Problem 2

Let $S$ be the solid obtained by rotating the region shown in the figure about the y-axis. Sketch a typical cylindrical shell and find its circumference and height. Use shells to find the volume of $S$. Do you think this method is preferable to slicing? Explain. Amrita B.

### Problem 3

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis.

$y = \sqrt{x}$ , $y = 0$ , $x = 1$ Amrita B.

### Problem 4

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis.

$y = x^3$ , $y = 0$ , $x = 1$ , $x = 2$ Amrita B.

### Problem 5

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis.

$y = e^{-x^2}$ , $y = 0$ , $x = 0$ , $x = 1$ Leon D.

### Problem 6

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis.

$y = 4x - x^2$ , $y = x$ Amrita B.

### Problem 7

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis.

$y = x^2$ , $y = 6x - 2x^2$ Amrita B.
Let $V$ be the volume of the solid obtained by rotating about the y-axis the region bounded by $y = \sqrt{x}$ and y = x^2 $. Find$ V $both by slicing and by cylindrical shells. In both cases draw a diagram to explain your method. Amrita B. Numerade Educator ### Problem 9 Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.$ xy = 1 $,$ x = 0 $,$ y = 1 $,$ y = 3 $ Amrita B. Numerade Educator ### Problem 10 Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.$ y = \sqrt{x} $,$ x = 0 $,$ y = 2 $ Amrita B. Numerade Educator ### Problem 11 Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.$ y = x^{\frac{3}{2}} $,$ y = 8 $,$ x = 0 $ Amrita B. Numerade Educator ### Problem 12 Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.$ x = -3y^2 + 12y - 9 $,$ x = 0 $ Amrita B. Numerade Educator ### Problem 13 Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.$ x = 1 + (y - 2)^2 $,$ x = 2 $ Amrita B. Numerade Educator ### Problem 14 Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.$ x + y = 4 $,$ x = y^2 - 4y + 4 $ Amrita B. Numerade Educator ### Problem 15 Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis.$ y = x^3 $,$ y = 8 $,$ x = 0 $; about$ x = 3 $ Amrita B. Numerade Educator ### Problem 16 Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis.$ y = 4 - 2x $,$ y = 0 $,$ x = 0 $; about$ x = -1 $ Amrita B. Numerade Educator ### Problem 17 Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis.$ y = 4x - x^2 $,$ y = 3 $; about$ x = 1 $ Amrita B. Numerade Educator ### Problem 18 Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis.$ y = \sqrt{x} $,$ x = 2y $; about$ x = 5 $ Amrita B. Numerade Educator ### Problem 19 Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis.$ x = 2y^2 $,$ y \ge 0 $,$ x = 2 $; about$ y = 2 $ Amrita B. Numerade Educator ### Problem 20 Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis.$ x = 2y^2 $,$ x = y^2 + 1 $; about$ y = -2 $ Amrita B. Numerade Educator ### Problem 21 (a) Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curve about the specified axis. (b) Use your calculator to evaluate the integral correct to five decimal places.$ y = xe^{-x} $,$ y = 0 $,$ x = 2 $; about the y-axis Amrita B. Numerade Educator ### Problem 22 (a) Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curve about the specified axis. (b) Use your calculator to evaluate the integral correct to five decimal places.$ y = \tan x $,$ y = 0 $,$ x = \frac{\pi}{4} $; about$ x = \frac{\pi}{2} $ Amrita B. Numerade Educator ### Problem 23 (a) Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curve about the specified axis. (b) Use your calculator to evaluate the integral correct to five decimal places.$ y = \cos^4 x $,$ y = -\cos^4 x $,$ \frac{-\pi}{2} \le x \le \frac{\pi}{2} $; about$ x = \pi $ Amrita B. Numerade Educator ### Problem 24 (a) Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curve about the specified axis. (b) Use your calculator to evaluate the integral correct to five decimal places.$ y = x $,$ y = \frac{2x}{(1 + x^3)} $; about$ x = -1 $ Amrita B. Numerade Educator ### Problem 25 (a) Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curve about the specified axis. (b) Use your calculator to evaluate the integral correct to five decimal places.$ x = \sqrt{\sin y} $,$ 0 \le y \le \pi $,$ x = 0 $; about$ y = 4 $ Amrita B. Numerade Educator ### Problem 26 (a) Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curve about the specified axis. (b) Use your calculator to evaluate the integral correct to five decimal places.$ x^2 - y^2 = 7 $,$ x = 4 $; about$ y = 5 $ Amrita B. Numerade Educator ### Problem 27 Use the Midpoint Rule with$ n = 5 $to estimate the volume obtained by rotating about the y-axis the region under the curve$ y = \sqrt{1 + x^3} $,$ 0 \le x \le 1 $. Amrita B. Numerade Educator ### Problem 28 If the region shown in the figure is rotated about the y-axis to form a solid, use the Midpoint Rule with$ n = 5 $to estimate the volume of the solid. Jacquelyn T. Numerade Educator ### Problem 29 Each integral represents the volume of a solid. Describe the solid.$ \displaystyle \int_{0}^3 2 \pi x^5 dx $ Amrita B. Numerade Educator ### Problem 30 Each integral represents the volume of a solid. Describe the solid.$ \displaystyle \int_{1}^3 2 \pi y \ln y dy $ Umar Sohail Q. Numerade Educator ### Problem 31 Each integral represents the volume of a solid. Describe the solid.$ 2 \pi \displaystyle \int_{1}^4 \frac{y + 2}{y^2} dy $ Amrita B. Numerade Educator ### Problem 32 Each integral represents the volume of a solid. Describe the solid.$ \displaystyle \int_{0}^1 2 \pi (2 - x)(3^x - 2^x) dx $ Amrita B. Numerade Educator ### Problem 33 Use a graph to estimate the x-coordinates of the points of intersection of the given curves. Then use this information and your calculator to estimate the volume of the solid obtained by rotating about the y-axis the region enclosed by these curves.$ y = x^2 - 2x $,$ y = \frac{x}{x^2 + 1} $ Amrita B. Numerade Educator ### Problem 34 Use a graph to estimate the x-coordinates of the points of intersection of the given curves. Then use this information and your calculator to estimate the volume of the solid obtained by rotating about the y-axis the region enclosed by these curves.$ y = e^{\sin x} $,$ y = x^2 - 4x + 5 $ Amrita B. Numerade Educator ### Problem 35 Use a computer algebra system to find the exact volume of the solid obtained by rotating the region bounded by the given curves about the specified line.$ y = \sin^2 x $,$ y = \sin^4 x $,$ 0 \le x \le \pi $; about$ x = \frac{\pi}{2} $ Amrita B. Numerade Educator ### Problem 36 Use a computer algebra system to find the exact volume of the solid obtained by rotating the region bounded by the given curves about the specified line.$ y = x^3 \sin x $,$ y = 0 $,$ 0 \le x \le \pi $; about$ x = -1 $ Amrita B. Numerade Educator ### Problem 37 The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method.$ y = -x^2 + 6x - 8 $,$ y = 0 $; about the y-axis Amrita B. Numerade Educator ### Problem 38 The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method.$ y = -x^2 + 6x - 8 $,$ y = 0 $; about the x-axis Amrita B. Numerade Educator ### Problem 39 The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method.$ y^2 - x^2 = 1 $,$ y = 2 $; about the x-axis WZ Wen Z. Numerade Educator ### Problem 40 The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method.$ y^2 - x^2 = 1 $,$ y = 2 $; about the y-axis WZ Wen Z. Numerade Educator ### Problem 41 The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method.$ x^2 + (y - 1)^2 = 1 $; about the y-axis Amrita B. Numerade Educator ### Problem 42 The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method.$ x = (y - 3)^2 $,$ x = 4 $; about$ y = 1 $ Amrita B. Numerade Educator ### Problem 43 The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method.$ x = (y - 1)^2 $,$ x - y = 1 $; about$ x = -1 $ Amrita B. Numerade Educator ### Problem 44 Let$ T $be the triangular region with vertices$ (0, 0) $,$ (1, 0) $, and$ (1, 2) $, and let$ V $be the volume of the solid generated when$ T $is rotated about the line$ x = a $, where$ a > 1 $. Express$ a $in terms of$ V $. Jacquelyn T. Numerade Educator ### Problem 45 Use cylindrical shells to find the volume of the solid.$ A $sphere of radius$ r $ Amrita B. Numerade Educator ### Problem 46 Use cylindrical shells to find the volume of the solid. The solid torus of Exercise 6.2.63 Amrita B. Numerade Educator ### Problem 47 Use cylindrical shells to find the volume of the solid.$ A $right circular cone with height$ h $and base radius$ r $ Amrita B. Numerade Educator ### Problem 48 Suppose you make napkin rings by drilling holes with different diameters through two wooden balls (which also have different diameters). You discover that both napkin rings have the same height$ h $, as shown in the figure. (a) Guess which ring has more wood in it. (b) Check your guess: Use cylindrical shells to compute the volume of a napkin ring created by drilling a hole with radius$ r $through the center of a sphere of radius$ R $and express the answer in terms of$ h \$. 