Page 100 - DMTH202_BASIC_MATHEMATICS_II

Unit 7: Definite Integral Applications

is there a symmetry in the graph of the curve between 0 and 2 and 0 and – 2, but the area under Notes
the curve from 0 to 2 is identical as the area under the curve from 0 to – 2. Thus, the complete area
among this curve and the x-axis is twice the area computation between x = 0 and x = 2. Observe
the lack of negative numbers in this solution.

2 3
3
  
2  x    2    4 12 4 16
A  2 0     2  dx  2 2x    2  2 2      2 4   2   
2 (1/2)x 
 
0 

 6 0    6    3 3 3 3
In performing area problems gaze for symmetry that will make the problem simpler and cut
down on the quantity of numbers you have to influence.
Example: An investment in a definite mining process is returning an total per month
that goes with the following formula
2
A  0.25x  14x  300
where x is the month and A is the amount for that month. This formula is applicable for 40 months.

What is the overall return for the first 12 months?
Solution:
The curve begins out at $300 for the zero month and drops off with time. In this problem the
th
height of a rectangle beneath the curve for every value of x displays the money obtained that
month and the sum of these rectangles is the total quantity for the period summed. For 12
months this amount is the area under the curve from the zero to the eleven month, or the
th
th
integral from 0 to 11.
11
2
T  0  0.25x  14x  300  dx
11
 x 3 x 3 
T   0.25  14  300x 
 3 2 0 
 11 3 11 2 
 
T   0.25  14  300 11 
 3 2 
T  $2564
Self Assessment

Fill in the blanks:

1. A straight line, y = const, above a distance in x is a ................................... .

2. The area of a rectangle is the height multiplied by the ................................... .
3. The area of a ...................................is (1/2)(sum of the opposite faces) (height).
4. The rectangles can be created in a numerous methods, inside the curve, outside the curve
or by using a ................................... .
5. It actually doesn’t make any variation how the rectangles are created since we are going to
take the limit by having their width to ................................... .
6. By means of a ................................... approach, and the knowledge that this outlines, or
integral, over a particular range in x is the area under the curve, A is the limit of the sum
as Dx leads to zero.

LOVELY PROFESSIONAL UNIVERSITY 95