   Chapter 10.2, Problem 57E

Chapter
Section
Textbook Problem

Set up an integral that represents the area of the surface obtained by rotating the given curve about the x-axis. Then use your calculator to find the surface area correct to four decimal places.57. x = t sin t, y = t cos t, 0 ≤ t ≤ π/2

To determine

To find: The surface area of the curve for the parametric equation x=tsint and y=tcost.

Explanation

Given:

The parametric equation for the variable x is as below.

x=tsint

The parametric equation for the variable y is as below.

y=tcost

The value t ranges from 0 to π2.

Calculation:

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

S=012πy(dxdt)2+(dydt)2dt

The value t ranges from 0 to π2.

Differentiate the variable x with respect to t.

x=tsintdxdt=tcost+sint

Differentiate the variable y with respect to t.

y=sin2tdydt=tsint+cost

Write the length of the curve formula.

S=0π22πy(dxdt)2+(dydt)2dt

Substitute (tcost+sint) for dxdt and (tsint+cost) for dydt in the above equation

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