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Chapter 5: Problem 45

What precalculus formula and representative element are used to develop the integration formula for the area of a surface of revolution?

Short Answer

Expert verified
The precalculus formula used to develop the integral formula for the area of a surface of rotation is the formula for the circumference of a circle, \(2蟺r\), and the representative element is an infinitesimally thin slice of the function with a width \(dx\), treated as a cylindrical element.

Step by step solution

01

Identify Precalculus Formula

The precalculus formula used is the formula for the circumference of a circle, which is \(2蟺r\), where \(r\) is the distance from the axis of rotation to the graph.
02

Identify Representative Element

The representative element in this case is an infinitesimally thin slice of the function width \(dx\) that can be treated as a small cylinder. The surface area of this slice is \(2蟺rh\), where \(r\) is the function \(f(x)\) at a given point and \(h\) is the slice's width, or \(dx\).
03

Combine to Form Integral

To find the total area of the surface of revolution, the surface areas of all the small slices must be added up. This is done by integrating the surface area formula \(\int 2蟺f(x)dx\)

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Most popular questions from this chapter

Think About It Consider the equation \(\frac{x^{2}}{9}+\frac{y^{2}}{4}=1 .\) (a) Use a graphing utility to graph the equation. (b) Set up the definite integral for finding the first quadrant arc length of the graph in part (a). (c) Compare the interval of integration in part (b) and the domain of the integrand. Is it possible to evaluate the definite integral? Is it possible to use Simpson's Rule to evaluate the definite integral? Explain. (You will learn how to evaluate this type of integral in Section \(6.7 .)\)

(a) use a graphing utility to graph the region bounded by the graphs of the equations, \((b)\) find the area of the region, and (c) use the integration capabilities of the graphing utility to verify your results. $$ y=x \sqrt{\frac{4-x}{4+x}}, \quad y=0, \quad x=4 $$

Find the length of the curve \(y^{2}=x^{3}\) from the origin to the point where the tangent makes an angle of \(45^{\circ}\) with the \(x\) -axis.

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Sketch the region bounded by the graphs of the functions, and find the area of the region. $$ f(x)=\cos x, \mathrm{~g}(x)=2-\cos x, 0 \leq x \leq 2 \pi $$
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