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Fundamental theorem of calculus review

Review your knowledge of the fundamental theorem of calculus and use it to solve problems.

What is the fundamental theorem of calculus?

The theorem has two versions.

a) ddxaxf(t)dt=f(x)

We start with a continuous function f and we define a new function for the area under the curve y=f(t):
What this version of the theorem says is that the derivative of F is f. In other words, F is an antiderivative of f. Thus, the theorem relates differential and integral calculus, and tells us how we can find the area under a curve using antidifferentiation.

b) abf(x)dx=F(b)F(a)

This version gives more direct instructions to finding the area under the curve y=f(x) between x=a and x=b. Simply find an antiderivative F and take F(b)F(a).
Want to learn more about the fundamental theorem of calculus? Check out this video.

Practice set 1: Applying the theorem

Problem 1.1
  • Your answer should be
  • an integer, like 6
  • an exact decimal, like 0.75
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4

Want to try more problems like this? Check out this exercise.

Practice set 2: Applying the theorem with chain rule

We can use the theorem in more hairy situations. Let's find, for example, the expression for ddx0x3sin(t)dt. Note that the interval is between 0 and x3, not x.
To help us, we define F(x)=0xsin(t)dt. According to the fundamental theorem of calculus, F(x)=sin(x).
It follows from our definition that 0x3sin(t)dt is F(x3), which means that ddx0x3sin(t)dt is ddxF(x3). Now we can use the chain rule:
Problem 2.1

Want to try more problems like this? Check out this exercise.

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