Application of Denite Integrals
Bridge Course, August 2012

Area enclosed by curves

Intuition

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

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

However, in general,

Example                                                                                                      MAXIMA CODE

Find the area bounded by a circle                                             

Figure 1:

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

Then we let , , , . Hence,

Exercises
1. Find the area bounded by the ellipse

Figure 2:

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

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

Figure 3: Question 5

Volume of Solids of Revolution

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

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                              

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

Radius of each circular disc:

Surface area of the each circular disc ():

Hence,

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

2. Derive the formula for the volume of Cylinder .