How do you use the Fundamental Theorem of Calculus to evaluate an integral?

Question

Emilia Aguilar · Accepted Answer

If we can find the antiderivative function #F(x)# of the integrand #f(x)#, then the definite integral #int_a^b f(x)dx# 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)=int_a^b v(t)dt#.  We know that integrating the velocity function gives us a position function.  So taking #s(b)-s(a)# results in a displacement.

Charles Arocha · Answer

undefined The Fundamental Theorem of Calculus states that if $ f $ is continuous on the closed interval $[a, b]$, and $ F $ is an antiderivative of $ f $ on $[a, b]$, then:

$$ \int_a^b f(x) \, dx = F(b) - F(a) $$

In simpler terms, to evaluate the integral of a function $ f $ over the interval $[a, b]$, you find an antiderivative $ F $ of $ f $, then evaluate $ F $ at $ b $ and subtract $ F $ evaluated at $ a $.