   Chapter 10.2, Problem 61E

Chapter
Section
Textbook Problem

# Find the exact area of the surface obtained by rotating the given curve about the x-axis.61. x = t3, y = t2, 0 ≤ t ≤ 1

To determine

To find: The surface area of the curve for the parametric equation x=t3 and y=t2.

Explanation

Given:

The parametric equation for the variable x is as follows.

x=t3

The parametric equation for the variable y is as follows.

y=t2

The value of t will range from 0 to 1.

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 1.

Differentiate the variable x with respect to t.

dxdt=3t2

Differentiate the variable y with respect to t.

dydt=2t

Write the length of the curve formula.

S=132πy(dxdt)2+(dydt)2dt

Substitute (3t2) for dxdt, (t2) for y, and (2t) for dydt in the above equation.

S=012πy(dxdt)2+(dydt)2dt=012π(t2)(3t2)2+(2t2)2dt=012πt29t2+4dtS=2π01t29t2+4dt

Convert the variable t inside the integration into variable u.

The expression (9t2+4) is u and 18tdt is du.

Convert the limits from variable t to variable u.

Substitute 0 for t in equation (9t2+4)=u.

(9t2+4)=u(9(0)2+4)=uu=4

Substitute 1 for t in equation (9t2+4)=u

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