Lecture in Finding Areas using Definite Integration

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
limitdefintgrl 1
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

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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:
area%20under%20curve%20formulaGraph
  area%20under%20curve%20formula
Example 1: Find the area between y = 7 – x2 and the x-axis between the values x = –1 and x = 2.
Example 2: Find the net area between y = sin x and the x-axis between the values x = 0 and x = 2π.

Area Between a Curve

The area between curves is given by the formulas below.

Formula 1: area%20between%20curves%20formula1
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.

eq0009M
Formula 2: area%20between%20curves%20formula2
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.


eq0010M
Example 1:1 Find the area between y = x and y = x2 from x = 0 to x = 1.

Example 2:1 Find the area between x = y + 3 and x = y2 from y = –1 to y = 1.

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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