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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
BuyFindarrow_forward

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Find the surface area generated by rotating the given curve about the y-axis.

65. x = 3t2, y = 2t3, 0 ≤ t ≤ 5

To determine

To find: The surface area of the curve for the parametric equation x=3t2 and y=2t3.

Explanation

Given:

The parametric equation for the variable x is as below.

x=3t2

The parametric equation for the variable y is as below.

y=2t3

The value t ranges from 0 to 5.

Calculation:

The parametric equation of the curve is rotated by y axis then the resulting surface is S=αβ2πx(dxdt)2+(dydt)2dt.

The value t will range from α to β.

The surface area of the curve is rotated by y axis.

S=052πx(dxdt)2+(dydt)2dt

Differentiate the variable x with respect to t.

dxdt=6t

Differentiate the variable y with respect to t:

dydt=6t2

Write the surface area of the curve is rotated by y axis.

S=052πx(dxdt)2+(dydt)2dt

Substitute (6t) for dxdt and (6t2) for dydt in the above equation.

S=052πx(dxdt)2+(dydt)2dt=052π(3t2)(6t)2+(6t2)2dt=0512πt236t2+36t4dt=2π×605t21+t2(2t)dt

Simplify further,

S=18π05t21+t2(2t)dt

Convert the variable t into variable u as below.

Substitute (1+t2)=u and 2tdt=du

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