Area Under a Graph

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      To find the area under the graph of a function f from a to b, we divide the interval [a,b] into n subintervals, all having the same length (b - a)/n. Observe the figure below.

Since f is continuous on each subinterval, f takes on a minimum value at some number c_i in each subinterval.
On can construct a rectangle with one side of length [x_{i - 1}, x_i], and the other side of length equal to the minimum distance f(c_i) from the x-axis to the graph of f.
The area of this rectangle is f(c_i) ¤x, where ¤x is (b - a)/n. The boundary of the region formed by the sum of these rectangles is called the inscribed rectangular polygon.
The area (A) under the graph of f from a to b follows below. Note that the summation sign Sigma is not an html character and will be denoted by £.
             _n
A = lim £ f(c_i) ¤x,      x_{i - 1} < c_i < x_i, where ¤x = (b - a)/n.
^{n-->infinity i = 1}
The area A under the graph may also be obtained by means of circumscribed rectangular polygons. In the case of the circumscribed polygons the maximum value of f on the interval [x_{i - 1}, x_i] is used.
Remember that the area obtained using circumscribed polygons should always be larger than that obtained by using inscribed rectangular polygons.

You may remember an easier way using the Fundamental Theorem of Calculus. That topic is also in the review section. Also notice the Applications of The Integral sub section as well.