How to evaluate definite integral without the fundamental theorem of calculus?

If we can find the antiderivative function

F

(

x

)

of the integrand

f

(

x

)

, then the definite integral

∫

b

a

f

(

x

)

d

x

can be determined by

F

(

b

)

-

F

(

a

)

provided that

f

(

x

)

is continuous.

We are usually given continuous functions, but if you want to be rigorous in your solutions, you should state that

f

(

x

)

is continuous and why.

FTC part 2 is a very powerful statement. Recall in the previous chapters, the definite integral was calculated from areas under the curve using Riemann sums. FTC part 2 just throws that all away. We just have to find the antiderivative and evaluate at the bounds! This is a lot less work.

For most students, the proof does give any intuition of why this works or is true. But let's look at

s

(

t

)

=

∫

b

a

v

(

t

)

d

t

. We know that integrating the velocity function gives us a position function. So taking

s

(

b

)

-

s

(

a

)

results in a displacement.