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

62. x = 2t2 + 1/t, y = 8 t , 1 ≤ t ≤ 3

To determine

To find: The surface area of the curve for the parametric equation x=2t2+1t and y=8t.

Explanation

Given:

The parametric equation for the variable x is as below.

x=2t2+1t

The parametric equation for the variable y is as below.

y=8t

The value t ranges from 1 to 3.

Calculation:

The surface area of the surface obtained by rotating the curve about the x axis is as below.

S=132πy(dxdt)2+(dydt)2dt

The value of t will range from 1 to 3.

Differentiate the variable x with respect to t.

dxdt=4t1t2

Differentiate the variable y with respect to t:

dydt=4t

Write the length of the curve formula.

S=132πy(dxdt)2+(dydt)2dt

Substitute (4t1t2) for dxdt and (4t) for dydt in the above equation.

S=132πy(dxdt)2+(dydt)2dt=132π(8t)(4t1t2)2+(4t)2dt=

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