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8 Application of Definite Integrals

Page history last edited by CAI, Sheng 11 years, 10 months ago

Application of Denite Integrals
Bridge Course, August 2012

 

Area enclosed by curves


Intuition

We know that the net (signed) area bounded by a curve Formula, Formula-axis, vertical lines Formula and Formula is Formula, but if we want to find the area bounded by two curves, say Formula and Formula, and two vertical lines, what should we do?

 

Denition
Let A be the area bounded by the curves Formula, Formula, and the lines Formula, Formula.
If Formula for all Formula in Formula, then

                                                      Formula

However, in general,

                                                     Formula

 

Example                                                                                                      MAXIMA CODE

Find the area bounded by a circle Formula                                            Formula

Figure 1: Formula

 

Let the area bounded by the circle be Formula. We can consider that Formula is bounded by two curve, Formulaand Formula. Since Formula for Formula, then

 

                                                     Formula

 

Then we let Formula, Formula, Formula, Formula. Hence,

 

                                                    Formula

 

Exercises
1. Find the area bounded by the ellipse Formula

Figure 2: Formula

2. Find the area bounded by the curve Formula, Formula and Formula, where Formula
3. Find the area bounded by the curve Formula and Formula

4. Find the area bounded by the curve Formula, Formula, Formula-axis and Formula, assume that Formula, Formula, Formula are positive real numbers.
5. Find the shaded area bounded by the curve Formula and Formula as shown in Figure 3

Figure 3: Question 5

 

Volume of Solids of Revolution


Denition
Let Formula be a function continuous on Formula. Then the volume Formula of a solid of revolution generated by revolving the region by the graph of Formula, Formula-axis, and the lines Formula, and Formula about the Formula-axis is

                                                   Formula

Note That: The meaning of the formula is that total volume equals to infinite sum of surface of individual element times its thickness

 

Example                                                                                                      MAXIMA CODE

1. Derive the formula for the volume of sphere Formula                             Formula

The volume of a sphere is the integral of infinitesimal circular discs of thickness Formula, assume that a sphere with center at the origin and radius Formula, then

Radius of each circular disc: Formula

Surface area of the each circular disc (Formula): Formula

Hence,

                                                  Formula

 

Exercises
1. Derive the formula for the volume of Cone Formula. The following graph may help you.


2. Derive the formula for the volume of Cylinder Formula.

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