Evaluating Definite Integrals Using the Fundamental Theorem

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  • 0:00 A Definite Integral
  • 0:34 The Fundamental…
  • 1:17 Example 1
  • 2:30 Example 2
  • 3:16 Lesson Summary
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Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

The fundamental theorem of calculus makes finding your definite integral almost a piece of cake. See how the definite integral becomes a subtraction problem after applying the fundamental theorem of calculus.

A Definite Integral

Let's get right into this lesson about definite integrals. A definite integral is the integral of a function from a starting point to an end point. You'll see little numbers at the top and bottom of your integral sign telling you where your integration begins and where it ends. The bottom value gives you your starting point and the top value gives you your end point.

This is an example of a general, definite integral:

definite integral

When you read it, you read it saying: 'the definite integral of the function f of x from point a to point b with respect to x.

The Fundamental Theorem of Calculus

To help us evaluate our definite integrals we have a fundamental theorem of calculus. This theorem tells us that the definite integral of f or x from point a to point b, with respect to x, is equal to the integral of f of x evaluated at point b minus the integral of f of x evaluated at point a.

definite integral

Mathematically we write this where our big F stands for the integral. This fundamental theorem of calculus actually helps us a lot when it comes to evaluating our definite integral. It actually makes it very easy for us to do, especially when we've memorized our common integrals as well as our common integration rules, such as the power rule.

Let's take a look at a couple of examples to see how easy it is.

Example 1

Evaluate:

definite integral

To evaluate this definite integral, we first find the integral of 3x^2, it is 3x^3 / 3 = x^3 + C. We have the constant of integration C when we take the general integral. When we take the definite integral we end up subtracting C from C, which essentially removes it, so we don't worry about it when we work with definite integrals.

Now we can apply the fundamental theorem of calculus. We first evaluate the x^3 + C at the point x = 3. Plugging in 3 for x we get: 3^3 + C = 27 +C.

Now we evaluate the x^3 + C at the point x = 1. Plugging in 1 for x we get: 1^3 + C = 1 +C.

Now subtracting the (1 + C) from the (27 + C) we get: (27 + C) - (1 + C) = 27 - 1 = 26.

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