   Chapter 10.3, Problem 68E

Chapter
Section
Textbook Problem

# Surface Area In Exercises 63-68, find the area of the surface generated by revolving the curve about each given axis. x = 1 3 r 3 , y = t + 1 , 1 ≤ t ≤ 2 , y-axis

To determine

To calculate: The surface area of the curve x=13t3,y=t+1 generated by revolving it about y-axis within the interval 1t2.

Explanation

Given:

The parametric equations,

x=13t3y=t+1

And, the interval 1t2.

Formula used:

The surface area of a smooth curve C given by x=f(t) and y=g(t) generated by revolving the curve C about the y-axis within the interval atb is given by formula:

S=2πabf(t)(dxdt)2+(dydt)2dt

Calculation:

Consider the equations,

x=13t3y=t+1

Differentiate x=13t3 with respect to ‘t’, to get,

dxdt=3t23=t2

Differentiate y=t+1 with respect to ‘t’, to get,

dydt=1

If smooth Curve C given by x=f(t) and y=g(t) does not cross itself on an interval atb then area S of surface of revolution formed by revolving the curve C about the

y-axis is given by formula:

S=2πabf(t)(d

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