### Finding Volumes Using the Method of Cylindrical ShellsThis section will explain how to find the volume generated by rotating a region around a given axis using the method of cylindrical shells. Please ensure to enter exact answers in terms of π (pi).#### Example Problems1. **Problem (a)** Given the function: \[ y = 3x - x^2, \, x = 0 \] Find the volume of the solid formed by rotating the region around the y-axis. \[ \text{Volume} = \underline{\hspace{80pt}} \]2. **Problem (b)** Given the function: \[ y = \frac{1}{x^2}, \, 1 \leq x \leq 3, \, y = 0 \] Find the volume of the solid formed by rotating the region around the y-axis. \[ \text{Volume} = \underline{\hspace{80pt}} \]### Explanation of the Method of Cylindrical ShellsThe method of cylindrical shells involves the following steps:1. **Identify the region to be rotated**: Define the boundaries of the region in terms of the given function(s).2. **Shell element**: Consider an infinitesimally thin vertical strip of width $ \Delta x $ within the region.3. **Radius of the shell**: Determine the distance of this strip from the axis of rotation.4. **Height of the shell**: Given by the function value at a specific $ x $-coordinate.5. **Volume of the shell**: The volume of the cylindrical shell is given by $ 2\pi(\text{radius})(\text{height})(\Delta x) $.6. **Integration**: Integrate this expression over the interval to find the total volume.This method is particularly useful when the strip is easier to describe vertically rather than horizontally.#### General Formula\[ V = \int_{a}^{b} 2\pi x f(x) \, dx \]Where:- $ f(x) $ is the function describing the height of the shell.- $ x $ is the distance from the y-axis to the shell.Remember to check the limits of integration according to the given bounds of $ x $.

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Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter8: Further Techniques And Applications Of Integration

Section8.3: Volume And Average Value

Problem 22E

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Suppose that r=12 cm and h=15 cm in the right circular cylinder. Find the exact and approximate a lateral area. b total area. c volume.
***Kindly solve the following completely within 20 minutes and draw the figure/graph for each, identifying method used: Circular Disc Method, Circular Ring Method, or Cylindrical Shell Method—Either Vertical element or Horizontal Element. 2. FIND THE VOLUME OF SOLID GENERATED BY REVOLVING THE AREA ENCLOSED BY: y= x^2+1, x=0, y=0, and x=2 about: (A)y=5
(b) Figure 1 shows the graph of y=÷x+- and y² = x+1. The shaded region A, B and C 3 3 are the regions bounded by the graph. y y? = x + 1 B 1 y = 3x+3 -2 Figure 1 Find the (i) total area of region A and C. (ii) volume of the solid generated by revolving region B about the x-axis.
The graphs of y = 7, y = g(x) (solid) and y = f(x) (dashed) are shown; the graph is not drawn to scale. Region One, = (5,7), B = (9, 7), C = (7, 2), and D = (1,0). The points labeled are A Select the expression(s) needed to find the volume of the solid of revolution formed when the region is revolved around the y-axis. Suppose Region One is revolved around the y-axis. xp 6 Jo 2m2(f(x)-g(2)) dr | 2ma(7- g(x)) da 5. 6.
(1 point) Book Problem 3 Use the method of cylindrical shells to find the volume V of the solid obtained by rotating the region bounded by the given curves about the y-axis: y = 50/x2 , y = 0, x = 2, x = 5; The circumference of a typical shell = and the height of this shell = Therefore the volume V of the solid = , dx = %3D
Step 1 The region is bounded by the graphs of the equations y sin x, y = 0, x=0, and xx $50 For the representative rectangle, the radius of the solid of revolution is R(x)=sin x According to the disk method, the volume of the solid of revolution, when the area is revolved about the x axis is -=["[RE] = Therefore, V-s Hence, v-=[" ( Step 2 Use the half angle identity for sin²x. sin²x1-cos 2x v=x["( IC V (sin(x)) -=[x-sin 2x] Submit Skip (you cannot come back) Let 1- cos 2x The volume of the solid of revolution is V Let Therefore, - -COS X For the above region, f(x)=sin x g(x)=0 2-0, D-11 ++ So, the area of the entire region is 7. Exercise (b) the centroid of the region bounded by the graphs of the equations Use integration by parts. 2 Step 1 The region is bounded by the graphs of the equations y sin x, y=0, x=0, and x 8. A d The area of the representative rectangle is dA= (sin.x Therefore, Using integration by parts y A.B X dx. Step 2 The x-coordinate of the centroid for a region of…
A region is enclosed by the equations below and contains the point (2,8). y = 8 sin y = 8(x- 2), y = 7x + 1 Find the volume of the solid obtained by rotating the region about the r-axis. Find the volume of the solid obtained by rotating the region about the y-axis.
Find the volumes of the solids generated by revolving the regions bounded by the graphs of the equations about the given lines. y = sqrt(x) y = 0 x = 3 (a) the x-axis (b) the y-axis (c) the line x=3 (d) the line x=9
Volume of solid of Revolution Problem Revolve about the (a) x-axis and (b) y-axis the area enclosed by y²=x and y=x²³.
Page < of 2 ZOOM 2 12. Which of the following is the (F) cross-section of this solid? hp 洲專劑

Question 5 All questions in this page are referring to the same solid. The volume of a solid is described by the double integral -8 (64 - x^2) ,2yaydk If the order of the integral is reversed, find the lower limit of the inner integral. Note: 1. sqrt refers to square root, e.g. sqrt(3) means v(3). 2. A refers to power, e.g. 213 means 2°. Question 6 With reference to Question 5, find the upper limit of the inner integral. o -sqrt(64 - y^2) O 8 O 64 O sgrt(64 - ya2)
Find the volumes of the solids generated by revolving the regions bounded by the graphs of the equations about the given lines. y = x2 y = 4x - x2 (a) the x-axis (b) the line y = 10

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