How do you find the area of the surface generated by rotating the curve about the y-axis y=x^2, 0 le x le 2?

Jovanny Ray

Jovanny Ray

Answered question

2023-02-12

How to find the area of the surface generated by rotating the curve about the y-axis y = x 2 , 0 x 2 ?

Answer & Explanation

Tyree Hayes

Tyree Hayes

Beginner2023-02-13Added 8 answers

Since we are rotating this solid around the y-axis, we are concerned with the x distance from the y-axis to the function. This relation is given by x = ± y . We're only dealing with positive x values, so we can reduce this to just x = y for our case.
The formula for the surface area of a solid generated by rotating some curve g(y) around the y-axis on y [ c , d ] is given by
A = 2 π c d g ( y ) 1 + ( g ( y ) ) 2 d y
We go from x = 0 to x = 2 , which is analogous to traveling from y = 0 to y = 4 , which is what we care about.
We will use g ( y ) = y . Note that g ( y ) = 1 2 y .
A = 2 π 0 4 y 1 + ( 1 2 y ) 2 d y
A = 2 π 0 4 y ( 1 + 1 4 y ) d y
A = 2 π 0 4 y + 1 4 d y
Let u = y + 1 4 . This implies that d u = d y . We'll also need to adjust the boundaries.
A = 2 π 1 / 4 17 / 4 u 1 2 d u
A = 2 π [ 2 3 u 3 2 ] 1 / 4 17 / 4
A = 4 π 3 ( ( 17 4 ) 3 2 - ( 1 4 ) 3 2 )
Note that 4 3 2 = 8 :
A = 4 π 3 ( 17 3 2 - 1 8 )
A = π 6 ( 17 17 - 1 )

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