A definite integral looks like this:
#int_a^b f(x) dx#
Definite integrals differ from indefinite integrals because of the #a# lower limit and #b# upper limits.
According to the first fundamental theorem of calculus, a definite integral can be evaluated if #f(x)# is continuous on [#a,b#] by:
#int_a^b f(x) dx =F(b)-F(a)#
If this notation is confusing, you can think of it in words as:
The integral of a function (#f(x)#) with limits #a# and #b# is the integral of that function evaluated at the upper limit (#F(b)#) minus the integral of that function evaluated at the lower limit (#F(a)#)
#F(x)# just denotes the integral of the function.
Note that you will get a number and not a function when evaluating definite integrals. Also, you have to check whether the integral is defined at the given interval.
Let's look at an example .
#int_-2^6 x^3+2 dx#
#x^3+2# is defined for all real numbers, so the boundaries of #a# and #b# are defined. To evaluate this definite integral, we first find the integral function and then plug in the upper limit of 6 into the integral function, and subtract the integral function evaluated at the lower limit of -2.
#int_-2^6 x^3+2 dx = [1/4x^4+2x ]_-2^6=(1/4(6)^4+2(6))-(1/4(-2)^4+2(-2))= (336)-(0)= 336#
