When you figure out definite integrals (which you can think of as a limit of Riemann sums), you might be aware of the fact that the definite integral is just the area under the curve between two points (upper and lower bounds. You are finding an antiderivative at the upper and lower limits of integration and taking the difference. The Fundamental Theorem of Calculus justifies this procedure.
The technical formula is:
The first part of the fundamental theorem stets that when solving indefinite integrals between two points a and b, just subtract the value of the integral at a from the value of the integral at b. The second part of the theorem gives an indefinite integral of a function. It looks complicated, but all it’s really telling you is how to find the area between two points on a graph.
Fundamental Theorem of Calculus Example
Example problem: Evaluate the following integral using the fundamental theorem of calculus:
Step 1: Evaluate the integral. This particular integral is evaluated using the integral rule for power functions:
∫x2 dx = ⅓ x3 = x3⁄3
Note: For an indefinite integral, you would normally include the +C; Here we’re ignoring it, as we want to find a specific area.
Step 2: Find the value of the integral at b, which is the value at the top of the integral sign in the problem. In this example, the value of “b” is 1, so:
x3⁄3 = 13⁄3 = 1⁄3.
Step 3: Find the value of the integral at a, which is the value at the bottom of the integral sign in the problem. In this example, the value of “a” is 0, so:
x3⁄3 = 03⁄3 = 0
Step 4: Subtract a (Step 3) from b (Step 2):
1⁄3-0 = 1⁄3
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Stephanie Glen. "Fundamental Theorem of Calculus: Simple Definition, Examples" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/integrals/fundamental-theorem-of-calculus/
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