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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.

66. x = ett, y = 4et/2, 0 ≤ t ≤ 1

To determine

To find: The surface area of the curve for the parametric equation ett and y=4et/2 .

Explanation

Given:

The parametric equation for the variable x is as below.

ett

The parametric equation for the variable y is as below.

y=4et/2

The limits for variable t will range between values 0 to 1 .

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 of t will range from α to β .

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

S=012πx(dxdt)2+(dydt)2dt

Differentiate the variable x with respect to t .

dxdt=et1

Differentiate the variable y with respect to t .

y=4et/2dydt=4et/212=2rt/2

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

S=012πx(dxdt)2+(dydt)2dt

Substitute (et1) for dxdt and (2rt/2) for dydt in the above equation.

S=012πx(dxdt)2+(dydt)2dt=012π(ett)(et1)2+(2rt/2)2dt=012π(ett)e2t+12et

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P-52RECh-10 P-53RECh-10 P-54RECh-10 P-55RECh-10 P-56RECh-10 P-57RECh-10 P-58RECh-10 P-1PCh-10 P-2PCh-10 P-3PCh-10 P-4PCh-10 P-5PCh-10 P-6P

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