Page 87 - DMTH202_BASIC_MATHEMATICS_II

Basic Mathematics-II

Notes As the lower integral and the upper integral are now appeared to be equivalent we discover that
f + g is integrable, and its integral equals the value of the lower integral, that is
b b b
a   f  g   x dx   I f   g  a  f   x dx  a  g   x dx ,
as preferred.

Alternate Theorem

This theorem can also be established as

c b c
a  f   x dx  a  f   x dx  b  f   x dx
Let us suppose we have a function f(x), and three points on the x axis: a, b and c:

We have that a < b < c. The definite integral

c
a  f   x dx
Provides us the area under the curve from a to c:

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