Mathematics, 21.02.2020 18:51 uberagentkenny

Let R and H be positive numbers. In questions (1) and (2), the region R is the triangular region determined by the vertices (0,0), (0, R) and (H, R) in the xy-plane. (1) Use the Washer Method to find the volume of the solid obtained when R is rotated about the x axis. (2) Use the Shell Method to find the volume of the solid obtained when R is rotated about the 3 axis. Is there a way to check the accuracy of the answers to questions (1) and (2) using the formula for the volume of a cone?

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Answer from: cadereymer24

V = 2*pi*R^2*H / 3 .... Accurate according to formula

Step-by-step explanation:

Given:

- Coordinates of the triangular region:

( 0 , 0 ) , ( 0 , R ) , ( H , R )

- The region is rotated about x-axis.

Find:

Use the Washer Method to find the volume of the solid obtained when R is rotated about the x axis.

Use the Shell Method to find the volume of the solid obtained when R is rotated about the x axis

Is there a way to check the accuracy of the answers to questions (1) and (2) using the formula for the volume of a cone?

Solution:

- Using washer method. Define functions f(x) and g(x), such that f(x) > g(x):

f(x) = R

g(x) = ( R / H )*x

- Set up the integral for washer method as follows:

$V = \pi *\int\limits^H_0 {[f(x)^2 - g(x)^2]} \, dx\\\\V = \pi *\int\limits^H_0 {[R^2 - (R/H)^2*x^2]} \, dx\\\\V = \pi {[R^2*x - \frac{R^2}{3*H^2} *x^3]} \limits^H_0\\\\V = \pi {[R^2*H - \frac{R^2*H}{3}]} \\\\V = \frac{2*\pi*R^2*H }{3}$

- Using the shell method we have to express the area A as a function of y first:

A(y) = 2*pi*Radius*Width

A(y) = 2*pi*R*H - 2*pi*y*(H-H*y/R)

A(y) = 2*pi*R*H - 2*pi*(H*y-H*y^2/R)

- Set up the integral and evaluate:

$V = \int\limits^a_b {A(y)} \, dy\\\\V = \int\limits^R_0 [{2*pi*y*H - 2*pi*(H*y -\frac{H*y^2}{R} )}] \, dy\\\\V = [{pi*H*y^2 - 2*pi*(H*y^2 / 2 -\frac{H*y^3}{R} ) }] \limits^R_0\\\\V = [{pi*R^2*H - 2*pi*(H*R^2 / 2 -\frac{H*R^3}{3R} )}] \\\\V = pi*R^2*H - \frac{1}{3}* \pi*R^2*H\\\\V = \frac{2*\pi * R^2 * H}{3}$

- The volume of the solid can be verified by removing a cone from a cylinder as follows:

V = V_cylinder - V_cone

V = pi*R^2*H - pi*R^2*H / 3

V = 2*pi*R^2*H / 3

- The volume of solid derived is accurate according to the formulas.

Answer from: Quest

x is greater than 5

step-by-step explanation:

2x+7 is greater than 17

-7 -7

2x is greater than 10

divide by 2 each side

x is greater than 5

Answer from: Quest

if you was gonna solve for a, a is a < -12

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Let R and H be positive numbers. In questions (1) and (2), the region R is the triangular region det...