Areas found by Integration 

Look at the above area bounded by the curve, x-axis, x = a, and x = b.  The area described is an irregular area which has no pat formula in geometry.  To find the area of the above enclosure, we can use integration.  If you recall, integration is a summation.  We will sum an infinite number of rectangles whose width approaches zero!!  The integral then approaches the exact area under the curve!  The following is a basic interpretation of the integral.

This implies to find the area, you need to know the right-hand limit, left-hand limit and the upper and lower limit.  The curve for the above figure has the equation f(x) = x² .  Thus to find the area, the integral would be

Sample problems!!
1)  Find the area bounded by x-axis, f(x) = x² - 3x and the lines x = 1 and x = 3.
Solution:
a)  Graph the region above!!

The area to integrate must be an enclosed area.  This time the upper bound is the x-axis, the lower bound is the curve and the right boundary is x = 3 and the left boundary is x = 1.  The picture looks like:

The integral and its solution is below:

Therefore, the area is 10/3 square units!

2)  Find the area bound by the curves f(x) = 2x² and g(x) = x + 1
Solution:  Graph it first!!

You need to calculate where these curves intersect to find the left and right boundary.  The line is the upper boundary and the parabola is the lower boundary.  To find the intersection points, set the two equations equal and solve:

Thus, the right boundary is x = 1 and the left boundary is x = -1/2!!
Here is the integral and its solution:

3)  Find the area between the x-axis and the graph of f(x) = x² -4 from
x = 0 to x = 3.
Solution:  Graph the equation and identify the area.

Notice that the area this time is in two parts.  The first area has the x-axis as the top boundary and the curve as the lower boundary.  The left limit is x = o and the right limit is x = 2.  The second area has the curve as the top boundary and the x-axis as the lower boundary.  The left limit is x = 2 and the right limit is x = 3.  You need two integrals to solve the problem.  The integrals and solution is:

Hope this was a help for you!!  The sample test is next!!  Have a great summer and be ready for an exciting year at college.  I wish all of you good luck and it was a pleasure being your instructor this year!!  You were a great bunch of students!!  Stay that way!!