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Lecture in Finding Areas using Definite Integration

(Last Updated On: December 8, 2017)
Finding Areas using Definite Integration
The area problem give us one of the interpretations of a definite integral that will lead us to the definition of the definite integral.

Let the Limit of n approaches to infinity (n → ∞)

Reviewing what you have learned in the topic Area Problem and the Riemann Sum series.  As long as f is continuous the value of the limit is independent of the sample points xi used. If we let the Limit of n approaches to infinity (n → ∞). The definite integral of f from a to b is the define
Lecture in Finding Areas using Definite Integration
provided the limit exists.  The definite integral is defined to be exactly the limit and summation that we looked at in the previous topic to find the net area between a function and the x-axis.

Area Under a Curve

The area between the graph of y = f(x)and the x-axis is given by the definite integral below. This formula gives a positive result for a graph above the x-axis, and a negative result for a graph below the x-axis.

Note: If the graph of y = f(x) is partly above and partly below the x-axis, the formula given below generates the net area. That is, the area above the axis minus the area below the axis.
Formula:
Lecture in Finding Areas using Definite Integration
  Lecture in Finding Areas using Definite Integration
Example 1: Find the area between y = 7 – x2 and the x-axis between the values x = –1 and x = 2.

Lecture in Finding Areas using Definite Integration
Lecture in Finding Areas using Definite Integration

Example 2: Find the net area between y = sin x and the x-axis between the values x = 0 and x = 2π.

Lecture in Finding Areas using Definite Integration
Lecture in Finding Areas using Definite Integration

Area Between a Curve

The area between curves is given by the formulas below.

Formula 1: Lecture in Finding Areas using Definite Integration
for a region bounded above and below by y = f(x) and y = g(x), and on the left and right by x = a and x = b.

Lecture in Finding Areas using Definite Integration

Formula 2: Lecture in Finding Areas using Definite Integration
for a region bounded left and right by x = f(y) and x = g(y), and above and below by y = c and y = d.


Lecture in Finding Areas using Definite Integration

Example 1:1 Find the area between y = x and y = x2 from x = 0 to x = 1.
Lecture in Finding Areas using Definite Integration
Lecture in Finding Areas using Definite Integration
Example 2:1 Find the area between x = y + 3 and x = y2 from y = –1 to y = 1.
Lecture in Finding Areas using Definite Integration

Lecture in Finding Areas using Definite Integration

Continue and apply what you have learned by answering some problems that you can find on this link Area Under a Curve and Between a Curve – Set 1 Problems.

credit: Bruce Simmons, Paul Dawkins (Lamar University)©2013 www.PinoyBIX.com

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