Using the Fundamental Theorem of Calculus to Show Antiderivatives

Instructor: Damien Howard

Damien has a master's degree in physics and has taught physics lab to college students.

In this lesson you will learn how the fundamental theorem of calculus allows us to represent definite integrals as antiderivatives. You'll then cement their relationship in your mind by working through an example problem.

Differential and Integral Calculus

So far in your studies of calculus you've been treating the subject as having two separate branches, differential calculus and integral calculus. In differential calculus you learned how to take derivatives and antiderivatives of polynomial, exponential, trigonometric, and logarithmic functions. In integral calculus you've begun working on definite integrals as limits of Riemann sums.


While not at all a comprehensive list, these are a few examples of common antiderivatives you should already know
antiderivative examples


While we've been treating differential and integral calculus separately up until now, it turns out the two are intrinsically linked together through something called the fundamental theorem of calculus. This theorem shows us that differentiation and integration are inverse processes, two separate sides of the same coin. In this lesson we're going to focus on how the fundamental theorem of calculus can be used to show antiderivatives.

The Fundamental Theorem of Calculus Part 2

The fundamental theorem of calculus can be split into two main parts. In order to see how this theorem ties in with antiderivatives we're going to focus our time on working with the second part of the theorem.


the fundamental theorem of calculus part 2


Up to this point when you wanted to evaluate a definite integral you would have to take the limit of a Riemann sum. Evaluating integrals this way can be long and tedious, but part 2 of the fundamental theorem of calculus tells us is that we can evaluate any definite integral of a continuous function from a to b by taking the antiderivative of that function at the lower and upper limits of the integral. In this way we can use the fundamental theorem of calculus to represent definite integrals as antiderivatives.

Antiderivative Example

To see how useful the second part of the fundamental theorem of calculus is, let's try working through the following example together. Find the area under the curve given by f(x) = x2/2 + x over the interval 1 to 2.


This graph shows the area under the curve created by our function f(x) over the interval 1 to 2.
graph showing area under the curve


Looking at our graph we can see the function here is continuous, i.e. there are no breaks in the curve. So we can find the area by solving the following definite integral.


example problem integral


Since we know the fundamental theorem of calculus we no longer need to use Riemann sums to solve this limit. We can start by taking the antiderivative of f(x)

x2/2 + x = x3/6 + x2/2 + C

Now, all we have to do to find the area under the curve is take the difference antiderivative evaluated at the integral's upper and lower limits, i.e. F(b) - F(a).

F(2) - F(1) = (23/6 + 22/2 + C) - (13/6 + 12/2 + C)

= 8/6 + 4/2 + C - 1/6 - 1/2 - C

= 8/6 - 1/6 + 4/2 - 1/2 + C - C

= 7/6 + 3/2

= 8/3

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