What is the volume of the solid generated when S is revolved about the line #y=3# where S is the region enclosed by the graphs of #y=2x# and #y=2x^2# and x is between [0,1]?

Answer 1

First let us determine where the two functions intersect

#2x^2=2x #

#2x^2-2x=0 #

#x(2x-2)=0 #

#x=0 # and #2x-2=0# so #x=1 # also

So the interval over which we will integrate is #0<=x<=1 #

I am going to use the method of washers to find the volume

Outer radius is #3-2x^2#

Inner radius is #3-2x #

Integral for volume is

#piint_0^1(3-x^2)^2-(3-2x)^2dx #

#piint_0^1[9-12x^2+4x^4]-[9-12x+4x^2]dx #

#piint_0^1 9-12x^2+4x^4-9+12x-4x^2dx #

#piint_0^1 4x^4-16x^2+12xdx #

#pi[4/5x^5-16/3x^3+6x^2] #

#pi[4/5-16/3+6-0] #

#pi[-68/15+6]=(22pi)/15 #

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Answer 2

To find the volume of the solid generated when region S is revolved about the line y=3, you can use the disk method.

First, find the points of intersection between the two curves y=2x and y=2x^2: 2x = 2x^2 x = 0 and x = 1

Now, integrate the area of the disks formed by revolving the region S around the line y=3 from x=0 to x=1: [V = \int_{0}^{1} \pi [(\text{outer radius})^2 - (\text{inner radius})^2] dx]

Outer radius is the distance from the line of revolution (y=3) to the outer curve (y=2x), which is (3 - 2x). Inner radius is the distance from the line of revolution to the inner curve (y=2x^2), which is (3 - 2x^2).

[V = \pi \int_{0}^{1} [(3 - 2x)^2 - (3 - 2x^2)^2] dx]

Calculate the integral to find the volume.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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