   Chapter 10.2, Problem 68E

Chapter
Section
Textbook Problem

Use Formula 1 to derive Formula 6 from Formula 8.2.5 for the case in which the curve can be represented in the form y = F(x), a ≤ x ≤ b.

To determine

To derive: The formula S=αβ2πx(dxdt)2+(dydt)2dt for the case y=F(x),axb.

Explanation

Given:

Write the parametric equation of the curve.

The parametric equation for the variable x.

x=f(t)

The parametric equation for the variable y.

y=g(t)

Here the variable t ranges from α to β.

Calculation:

Rotating the curve y=F(x),αxβ about x axis, the surface area of the surface obtained is,

S=ab2πy1+(dydx)2dx (1)

Write the chain rule for dydx.

dydx=dydtdxdt

if dxdt0

Substitute (dydtdxdt) for (dydx) and (dx) for (dxdtdt) in equation (1).

By substitution rule a=x(α) and b=x(β).

S=αβ2πy1+(dydx)2dx=αβ2πy1+[(dydt)2(dxdt)2]dxdtdt=αβ2πy[(dx<

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