   Chapter 8, Problem 10RE

Chapter
Section
Textbook Problem

Find the area of the surface obtained by rotating the curve in Exercise 9 about the y-axis.

To determine

To find: The area of the surface obtained by rotating the curve about y-axis.

Explanation

Given information:

The function of the curve is y=1x(t1)dt1x16 (1)

The lower limit is 1 and the upper limit is 16.

The Fundamental Theorem of Calculus, Part 1 is shown below:

g(x)=axf(t)dtaxb

Here, the continuous function on the interval [a,b] is f and the function is g.

Condition for the theorem to be valid:

• If the function is continuous on the interval [a,b] and differentiable on (a,b) .
• Also for the continuous condition, g'(x)=f(x) .

Calculation:

Apply Fundamental Theorem of Calculus (Part 1) in Equation (1).

y'(x)=x1dydx=x1

The expression to find the area of the surface (S) obtained by rotating the curve about the y-axis is shown below:

S=ab2πx1+(dydx)2dx (2)

Here, the derivative of the function y is dydx , the lower limit is a, and the upper limit is b.

Substitute x1 for dydx , 1 for a, and 16 for b in Equation (2)

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