Question

Find the exact area of the surface obtained by rotating the curve about the x-axis.

Answer 1

\[4x=y^2+8, 5 \le x \le 10\]

Answer 2

\[x = \frac{ 1 }{ 4 }y^2+2, x' = \frac{ 1 }{ 2 }y\]\[S = \int\limits_{5}^{10}2piyds = 2\pi \int\limits_{5}^{10}y \sqrt{1+(x')^2}dy=2\pi \int\limits_{5}^{10}y \sqrt{1+\frac{ 1 }{ 4 }y^2}dy\]\[u = 1+ (1/4)y^2, du = (1/2)ydy\]Changing limits:\[u = 1 +(1/4)25 = 29/4, u = 1+(1/4)100 = 26\]\[S = 2\pi \int\limits_{29/4}^{26}\sqrt{u}(2du)\]

Answer 3

\[= 4\pi \int\limits_{29/4}^{26}u^{1/2}du = 4\pi [ \frac{ 2 }{ 3 }u^{3/2}]_{29/4}^{26}\]\[\frac{ 8\pi }{ 3 }[usqrt{u}]_{29/4}^{26}\]\[=\frac{ 8\pi }{ 3 }[26\sqrt{26}-\frac{ 29 }{ 4 }\sqrt{\frac{ 29 }{ 4 }}]\]

Answer 4

So yea, I need to find the exact area and not sure how to continue from the last step.

Answer 5

that is the exact area?

Answer 6

It wants the answer like
S=49π (an example)
or whatever, I blame the integration numbers for giving me weird numbers haha.

Answer 7

I'm not sure if my work is wrong or not, maybe that is why my last step is maybe weird?