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Above we have a static version of the Airplane Wing applet. As you saw in the applet and the lab exercises, the applet can be used to approximate the volume of the given airplane wing, or the amount of fuel contained. The mathematical process being applied is calculating volumes by slicing or cross-sections. How does this tie in with the definite integral?

(* General method and example taken from Chapter 6, Section 6.2, Calculus by Stewart)

Step 1. Break up Q into a large number of small parts.

Step 2. Approximate each small part by a quantity of the form .

Step 3. Approximate Q by a Riemann sum.

Step 4. Take the limit and express Q as a definite integral.

We can summarize the above as the following definition:

Let Q be a solid that lies between x = a and x = b. If the cross-sectional area of Q in a plane through x and perpendicular to the x-axis is A(x), where A is a continuous function, then the volume of Q is:

We now apply this method to the calculation of a well-known formula, the volume inside a sphere with radius, r.

We show that this volume is given by   V = 4/3 p r³.

Solution: Let's assume that the sphere is centered at the origin. A vertical cross-section at the origin intersects the sphere in a circle whose radius (from the Pythagorean Theorem) is . Thus, the cross-sectional area is

 

A (x) = p y² = p ( r² - x² )

 

Using the definition of volume with a = -r and b = r, we have

 

 

 

=

 

=

= 4/3 p r³ .

 

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Comments to: Professor Carol G. Crawford, at cgc@nadn.navy.mil or Professor Mark D. Meyerson, at mdm@nadn.navy.mil