This document discusses several applications of integration including: 1. Finding the area between curves using definite integrals. Examples are given for finding areas bounded by various functions. 2. Using integrals to calculate volumes, such as approximating the volume of a pyramid with boxes. 3. Using integrals in statistics, such as defining probability density functions, and calculating mean, mode and median which require integration. Integrals are also widely used in physics, mathematics and engineering.

1. Applications of Integration
1. Area between curves:
We have seen how integration can be used to find an area between a curve and the x-axis.
With very little change we can find some areas between curves; indeed, the area between
a curve and the x-axis may be interpreted as the area between the curve and a second
"curve'' with equation y=0. In the simplest of cases, the idea is quite easy to understand.
Example 1: Find the area of the region bounded by y = 2x, y = 0, x = 0 and x = 2
Solution:
We shall now use definite integrals to find the area defined above. If we let f(x) = 2x, using
the formula of the area given by the definite integral above, we
Example 2: Find the area of the region bounded by y = 0.1 x3, y = 0, x = 2 and x = 4.
Solution:
We first graph the given function and identify the region whose area is to be found.

2. Use the definite integrals to find the area as follows:
4
4 4 4
3 3
2 2 2
(0.1 ) 0.1 0.1
4
x
x dx x dx
 
   
 
  =
4 4
2
4 2
0.1 6
4 4
unit
 
 
 
 
Example 3: Find k so that the area of the finite region bounded by the curve of y = - x( x
- k) and the x axis is equal to 4/3 units2.
Solution: The graph of the given function is a parabola that opens downward and has two x
intercepts: x = 0 and x = k. The finite region bounded by the curve and the x axis is limited at
the x intercepts as shown in the graph below.
The area between the curve and y = 0 is given by
Expand - x( x - k)
As expected, the expression for the area includes the parameter k which is calculated by setting
the area equal to 4/3. Hence
Solve the above equation for k to obtain k=2
2. Volume
Example: Find the volume of a pyramid with a square base that is 20 meters tall and 20 meters
on a side at the base. As with most of our applications of integration, we begin by asking how
we might approximate the volume. Since we can easily compute the volume of a rectangular
prism (that is, a "box''), we will use some boxes to approximate the volume of the pyramid, as
shown in below figure on the left is a cross-sectional view, on the right is a 3D view of part of
the pyramid with some of the boxes used to approximate the volume.

3. Solution:
3. Statistics
In case of probability, there are a huge application of integration.
For defining the probability density function of a continuous distribution (like normal
distribution) need integration. Finding of mean, mode and median also required integration
For standard normal distribution,

4. Besides this, integration are mostly used in physics, mathematics and engineering
sciences. Some of useful application areas are
a. Distance, Velocity, Acceleration
b. Average value of a function
c. Center of Mass
d. Kinetic energy; improper integrals
e. Surface Area