Applications Of Integrals

(i) Area of region f(x) bounded between x =a to x = b is
=Sum of areas of rectangle shown in shaded part
(ii) If f{x)>g(x) when defined in [a,b], then

Description of Situation As the given curve
cannot be integrated, thus we have to bound this function by
using above mentioned concept.
Graph for
Since,



Also, Area of two rectangles

Given Functions:

The functions and are defined as:

and

Step 1: Understanding the Region of Interest

The region of interest is defined by:

Step 2: Comparison of and

We need to find the minimum of and over the interval . We first calculate these values:

• For , , and .
• At , we find and . So, the minimum value is at .
• For values of in , we compute the values for and and determine the minimum of the two functions at each point.

Step 3: Setting up the Integral for the Area

The area is the integral of the minimum of and from to :

Step 4: Calculating the Area

After performing the integration, we find that the area of the region is:

Step 5: Final Answer

The value of is:

The final value of is 6.