Fundamental theorem of calculus
Objective. Connect accumulation, area and antiderivatives.
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- 2. Worked example
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Learn: the key idea
The fundamental theorem of calculus links differentiation and integration: Part 1 states that if F(x)=∫[a,x] f(t)dt, then F'(x)=f(x), so accumulation functions have derivative equal to the original integrand; Part 2 states that ∫[a,b] f(x)dx = F(b)−F(a) for any antiderivative F of f. This means definite integrals can be evaluated by finding an antiderivative rather than computing limits of Riemann sums directly. For ∫₁³ 2x dx, an antiderivative is F(x)=x², so the integral equals F(3)−F(1)=9−1=8. The theorem justifies why the area function and antiderivatives are essentially the same object up to a constant.
Worked example
Apply the fundamental theorem of calculus to evaluate ∫₁³ 2x dx.
- 1Find an antiderivative of 2x: F(x)=x².
- 2By the fundamental theorem, ∫₁³2x dx = F(3)−F(1).
- 3Compute F(3)=9 and F(1)=1.
- 4Subtract: 9−1=8, giving the value of the integral.
Answer: 8
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Start practisingCommon mistake. Students forget to evaluate F at both limits and subtract, instead just writing the antiderivative formula as if it were the final numeric answer, omitting the crucial F(b)−F(a) step.
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