The Two Main Antiderivative Rules
Special Case Antiderivatives (The ones you have to memorize)
U-Substitution
Initial Value Problems: Finding the +C
Fundamental Theorem of Calculus Part 1

The Actual Fundamental Theorem of Calculus Part 2

Now that you understand how the Definite Integral was developed, you are given the Fundamental Theorem of Calculus Part 2, which lays out the basic mechanics of the Definite Integral process.

Definition: Fundamental Theorem of Calculus Part 2 = Definite Integral

If you have a continuous function, f ( x ) , on a closed x-interval [ a , b ], then the result of the antiderivative will be F ( b ) F ( a ) .

a b f ( x ) dx = F ( x ) | a b = F ( b ) F ( a )

What the notation is directing you to do is to; 1) take the antiderivative ( undo the derivative) of your given equation, and then 2) evaluate by plugging the upper limit , b ,into the result and subtracting the lower limit , a ,plugged into the result.

I always say Top Bottom . Plug the top value in and subtract the bottom value plugged in. Whenever in doubt, math rules generally work under that top minus bottom rule.

The answer to a Definite Integral represents the Net Area between the curve f ( x ) and the x-axis on a closed x-interval [ a , b ], a definite value.

Note: Since the Fundamental Theorem of Calculus Part 2 has a specific x-interval [ a , b ], this means that the lower bound, a ,will always be the smaller x-value ,and the upper bound, b ,will always be the larger x-value .

If your integral’s bounds are not setup with the lower bound as the smaller x-value and the upper bound as the larger x-value , then you will have to flip the bounds by performing a rewrite on the definite integral. To flip the bounds, you put a negative sign () in front of the integral.

a b f ( x ) dx = b a f ( x ) dx

5 3 f ( x ) dx = 3 5 f ( x ) dx

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