Question

Does the following make sense? f(x)=xsqrt(8-2x^2) , 0= 13 years ago

Answer 1

Given \[f(x)=x \sqrt{8-2x^2} , 0\le x \le2\]
for a graph


find surface area?

i did it like that:

\[f(x)=x \sqrt{8-2x^2} = \frac{ 1 }{ 2 } \int\limits_{0}^{2}2x \sqrt{8-2x^2}\]

\[u=8-2x^2\]
du=2xdx
x=0 ---> u=8-2*0^2=8
x=2 ---> u=8-2*2^2=0

follows:
\[\frac{ 1 }{ 2 }\int\limits_{8}^{0}\sqrt{u}du=\frac{ 1 }{ 2 }\int\limits_{8}^{0}u ^{\frac{ 1 }{ 2 }}du=\left[ \frac{ 1 }{ 3 } u ^{\frac{ 3 }{ 2 }}\right]_{8}^{0}=\frac{ 1 }{ 3 }*0-\frac{ 1 }{ 3 }*8^{\frac{ 3 }{ 2 }}=\frac{ 16\sqrt{2} }{ 3 }\]

so answer is (16*2^1/2)/3?