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 .

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