   Chapter 10.3, Problem 62E

Chapter
Section
Textbook Problem

# Surface Area In Exercises 61–64, write an integral that represents the area of the surface generated by revolving the curve about the x-axis. Use a graphing utility to approximate the integral.Parametric Equations Interval x = 1 4 t 2 ,         y = t + 3 0 ≤ t ≤ 3

To determine

To calculate: Area of surface generated by revolving the curve with parametric equations x=14t2 and y=t+3 on the interval 0t3.

Explanation

Given:

The provided parametric equations are:

x=14t2 and y=t+3

And the provided interval is:

0t3

Formula used:

If a smooth curve C given by x=f(t) and y=g(t) does not cross itself on an interval atb, then the area S of the surface of revolution about the x-axis is given by:

S=2πabg(t)(dxdt)2+(dydt)2dt

Calculation:

Consider the provided parametric equations:

x=14t2 and y=t+3

Now, differentiate x with respect to t, to get:

dxdt=12t …… (1)

And, differentiate y with respect to t, to get:

dydt=1 …… (2)

Now, If a smooth curve C given by x=f(t) and y=g(t) does not cross itself on an interval atb, then the area S of the surface of revolution about the x-axis is given by:

S=2πabg(t)(dxdt)2+(dydt)

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