Question #184946

Determine the surface area generated by revolving the curve y=1/3 x³ over the interval (0,2), about the x-axis.


1
Expert's answer
2021-05-07T08:13:15-0400

Solution

If the curve y = f(x), a≤x≤b is rotated about the x-axis, then the surface area is given by

A=2πabf(x)1+[f(x)]2dxA=2\pi\int_a^bf(x)\sqrt{1+[f'(x)]^2}dx

In this case f(x) = 1/3 x³, f’(x) = x2 ,a = 0, b = 2 

A=2π0213x31+x4dxA=2\pi\int_0^2\frac{1}{3}x^3\sqrt{1+x^4}dx

Substitution y = x4 

A=π60161+ydy=π9(1+y)3/2016=π9(173/21)=24.118A=\frac{\pi}{6}\int_0^{16}\sqrt{1+y}dy=\frac{\pi}{9}(1+y)^{3/2}|_0^{16}=\frac{\pi}{9}(17^{3/2}-1)=24.118

Answer

A=π9(173/21)=24.118A=\frac{\pi}{9}(17^{3/2}-1)=24.118



Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!
LATEST TUTORIALS
APPROVED BY CLIENTS