Find the area of the surface generated by revolving the curve about each given axis.x = 6t, y = 9t,0sts3(a)х-ахis(b)у-аxisNeed Help?Read ItWatch It

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Trigonometry (MindTap Course List)

10th Edition

ISBN:9781337278461

Author:Ron Larson

Publisher:Ron Larson

Chapter2: Analytic Trigonometry

Section2.3: Solving Trigonometric Equations

Problem 11ECP

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Find the area of the surface generated by revolving the curve about each given axis.
x = 6t, y = 9t,
0sts3
(a)
х-ахis
(b)
у-аxis
Need Help?
Read It
Watch It

Expert Solution

Step 1

Here, we need to find the surface generated by revolving the curve $x = 6 t, y = 9 t, 0 \leq t \leq 3,$ about each given axis.

For (a),

Now, $x' (t) = 6 and y' (t) = 9 .$

The area S of the surface of revolution formed by revolving about the x-axis is given by,

$S_{x} = 2 π \int_{0}^{3} y (t) \sqrt{{(\frac{dx}{dt})}^{2} + {(\frac{dy}{dt})}^{2}} d t = 2 π \int_{0}^{3} (9 t) \sqrt{{(6)}^{2} + {(9)}^{2}} d t = 18 π \int_{0}^{3} 3 \sqrt{13} t d t = 243 \sqrt{13} π .$

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