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 2 years ago
Find the area obtained from obtained by rotating the curve about the axis
\[9x=y^2+18\]
\[2 \le x \le 6\]
 2 years ago
Find the area obtained from obtained by rotating the curve about the axis \[9x=y^2+18\] \[2 \le x \le 6\]

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Eyad
 2 years ago
Best ResponseYou've already chosen the best response.1Write the curve as x = y^2/9 + 2. If x = 2, then y = 0 If x = 6, then y = 6 (the rotation of this region takes care of the negative counterpart.) A = integral 2 pi y sqrt(1 + [f '(y)]^2) dy = (2 pi) * integral(y = 0 to 6) y sqrt(1 + [2y/9]^2) dy = (2 pi/9) * integral(y = 0 to 6) y sqrt(81 + 4y^2) dy = (2 pi/9) * [(1/8) * (2/3)(81 + 4y^2)^(3/2) for y = 0 to 6] = 98 pi / 3.

MathSofiya
 2 years ago
Best ResponseYou've already chosen the best response.0\[A = \int 2 \pi y \sqrt{(1 + [f '(y)]^2)} dy\] \[= (2 \pi) * \int_{0}^{6} y \sqrt{(1 + [2y/9]^2)}dy\] \[= \frac{2\pi}{9}* \int_{0}^{6} y \sqrt{(81 + 4y^2)} dy\] \[= \frac{2\pi}{9} * \left[\frac18 * (\frac23)(81 + 4y^2)^{\frac32} \right] _{0}^{6}= \frac{98\pi}{3}\]

MathSofiya
 2 years ago
Best ResponseYou've already chosen the best response.0how did we go from line 2 to 3?

KingGeorge
 2 years ago
Best ResponseYou've already chosen the best response.0We're rotating around the xaxis correct?

KingGeorge
 2 years ago
Best ResponseYou've already chosen the best response.0In that case, you have \[y^2=9x18\implies y=\pm3\sqrt{x2}\]However, since we're looking for a full rotation, we throw out the negative. We want to integrate \[\large 3\pi\int_2^6\left(\sqrt{x2}\right)^2dx\]

MathSofiya
 2 years ago
Best ResponseYou've already chosen the best response.0oh shoot I'll do it again

KingGeorge
 2 years ago
Best ResponseYou've already chosen the best response.0I think eyad might have been rotating around the yaxis and finding surface area.

KingGeorge
 2 years ago
Best ResponseYou've already chosen the best response.0Wait, we want surface area and not volume.

KingGeorge
 2 years ago
Best ResponseYou've already chosen the best response.0In that case, we should be integrating \[\large 6\pi\int_2^6\left(\sqrt{x2}\right)dx\]

MathSofiya
 2 years ago
Best ResponseYou've already chosen the best response.0I think you're right

KingGeorge
 2 years ago
Best ResponseYou've already chosen the best response.0\[2\pi r\]But we can factor a 3 out of the r.

MathSofiya
 2 years ago
Best ResponseYou've already chosen the best response.0oh ok...lets see if I can do the integral

MathSofiya
 2 years ago
Best ResponseYou've already chosen the best response.0with my latex skills it's gonna be about 15 min just so you know

KingGeorge
 2 years ago
Best ResponseYou've already chosen the best response.0I've got plenty of time.

MathSofiya
 2 years ago
Best ResponseYou've already chosen the best response.0\[9x=y^2+18\] \[y=\sqrt{9x18}\] \[\frac{dy}{dx}=\frac 92 \left(9x18\right)^{\frac 12}\] \[S= \int_{2}^{6} 2 \pi x \sqrt{(1 + [f '(x)]^2)} dx\] \[S= \int_{2}^{6} 2 \pi x \sqrt{\left(1 + \left[\frac 92 \left(9x18\right)^{\frac 12}\right]^2\right)} dx\] \[S= 2 \pi\int_{2}^{6} x \sqrt{\left(1 + \left[\frac 92 \left(9x18\right)^{\frac 12}\right]^2\right)} dx\]

MathSofiya
 2 years ago
Best ResponseYou've already chosen the best response.0still working on it...

KingGeorge
 2 years ago
Best ResponseYou've already chosen the best response.0You're doing this using arc length?

KingGeorge
 2 years ago
Best ResponseYou've already chosen the best response.0I would not recommend that method.

MathSofiya
 2 years ago
Best ResponseYou've already chosen the best response.0No I thought that was for surface area

MathSofiya
 2 years ago
Best ResponseYou've already chosen the best response.0what method would you recommend?

KingGeorge
 2 years ago
Best ResponseYou've already chosen the best response.0Just use the formula for circumference of a circle. \(C=2\pi r\). Where \(r\) is your function solved for \(y\). Integrate from 2 to 6, and bam! There's the answer.

MathSofiya
 2 years ago
Best ResponseYou've already chosen the best response.0I'm soo silly ...sorry

KingGeorge
 2 years ago
Best ResponseYou've already chosen the best response.0We would only solve for x if we're revolving around the yaxis.
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