Practice Fundamental theorem of calculus
The fundamental theorem of calculus connects integration with differentiation.
A definite integral can be evaluated using the difference of an antiderivative at its endpoints.
Exercises with solutions
The fundamental theorem of calculus links the two areas: if F is an antiderivative of f, then the integral from a to b of f(x) dx equals F(b) − F(a). So you integrate using an antiderivative and substitute the limits.
Exercise 1
Evaluate the integral of 2x from 1 to 3.
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8
An antiderivative is x². Substituting: 3² − 1² = 9 − 1 = 8.
Exercise 2
Evaluate the integral of 3x² from 0 to 2.
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8
An antiderivative is x³. Substituting: 2³ − 0³ = 8 − 0 = 8.
Exercise 3
What does the theorem give when the upper and lower limits are equal?
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0
Then F(a) − F(a) = 0. An integral over an interval of length zero is always zero.
Exercise 4
Evaluate the integral of x from 0 to 4.
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8
An antiderivative is x²/2. Substituting: 16/2 − 0/2 = 8.
Common mistake
Swapping the limits. The theorem requires upper minus lower limit; the other way round you get the negative result. And overlooking that areas below the x-axis count as negative.
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