Finding the volume by Shell method: y = x 2 , y = 2 −...

Nash Frank

Nash Frank

Answered

2022-07-16

Finding the volume by Shell method: y = x 2 , y = 2 x 2 , y = x 2 , y = 2 x 2 about the line x = 1.
what I get from this after graphing is:
2 π ( 1 x ) ( 2 2 x 2 ) d x
which becomes: 2 π ( 2 2 x 2 2 x + 2 x 3 ) d x
integrating that I get:
2 π [ 2 x 2 3 x 3 x 2 + 1 2 x 4 ] from 0 to 1
My answer is 5 3 π but my book says the answer is: 16 3 π
Could someone tell me where I went wrong? Was it the upper lower bounds?

Answer & Explanation

Monastro3n

Monastro3n

Expert

2022-07-17Added 15 answers

Explanation:
The limits of the integration is given by where the lines cross each other:
x 2 = 2 x 2 x = ± 1
so the limits shold be from -1 to 1.
Haley Madden

Haley Madden

Expert

2022-07-18Added 7 answers

Explanation:
The region in the xy-plane bounded by the graphs y = x 2 and y = 2 x 2 is symmetric about the y-axis, so the projection of this region onto the x-axis should be a symmetric interval of the form [-a,a]. In this case the points of intersection of the graphs have x-coordinates x = ± 1. Thus, your limits of integration should really be from x = 1 to x = 1, not 0 to 1.

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