   Chapter 10.2, Problem 66E

Chapter
Section
Textbook Problem

# Find the surface area generated by rotating the given curve about the y-axis.66. x = et − t, 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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