Integrals · Definition

Fundamental theorem of calculus

calculusintegralstheorems

Before you start you need to know

The fundamental theorem of calculus unites differentiation and integration, which are inverse operations. Thanks to it, we can compute areas without summing rectangles.

Definition

It links derivatives and integrals: a definite integral equals an antiderivative evaluated at the endpoints, and differentiating an integral with variable upper limit returns the integrand.

\[\begin{gathered}\int_a^b f(x)\,dx=F(b)-F(a)\\ \frac{d}{dx}\int_a^x f(t)\,dt=f(x)\end{gathered}\]

This is the result that joins the two halves of calculus: differentiation and integration are inverse operations. It has two parts. The first says \(\frac{d}{dx}\int_a^x f(t)\,dt=f(x)\); the second, that \(\int_a^b f(x)\,dx=F(b)-F(a)\) for any antiderivative \(F\).

Thanks to the second part we no longer add up rectangles: we just find an antiderivative and evaluate it at the endpoints.

It is perhaps the most important theorem of Calculus I: it explains why accumulated area and rate of change are two faces of the same thing.

Think about it

Compute \(\int_0^2 x\,dx\).

An antiderivative is \(x^2/2\); it gives \(2-0=2\).

If \(F(x)=\int_a^x f\) and \(f\) has a jump at \(c\), is \(F\) differentiable at \(c\)?

No: \(F\) is continuous, but its one-sided slopes at \(c\) are the two different values of \(f\), so it has a corner.

Exercises 20

Compute the following limits

  1. \(\displaystyle \lim_{x\to0}\frac1x\int_0^x\cos(t^2)\,dt\)

Compute the following integrals

  1. \(\displaystyle \int_{0}^{2} \left(3 x^{2} + 1\right)\,dx\)
  2. \(\displaystyle \int_{1}^{3} 2 x\,dx\)
  3. \(\displaystyle \int_{0}^{\pi} \sin {x}\,dx\)
  4. \(\displaystyle \int_{0}^{1} e^{x}\,dx\)
  5. \(\displaystyle \int_{1}^{e} \frac{1}{x}\,dx\)
  6. \(\displaystyle \int_{0}^{4} \sqrt{x}\,dx\)
  7. \(\displaystyle \frac{d}{dx}\int_0^x t^2\,dt\)
  8. \(\displaystyle \int_1^3 2x\,dx\)
  9. \(\displaystyle \frac{d}{dx}\int_0^{x^2}e^{t}\,dt\)
  10. \(\displaystyle \int_{-1}^{2}x^2\,dx\)

Problems

  1. According to the fundamental theorem of calculus, \(\dfrac{d}{dx}\displaystyle\int_{1}^{x} \cos(t^{2})\,dt\) equals...
  2. What does the second part of the fundamental theorem of calculus state?
  3. By the fundamental theorem, what is \(\dfrac{d}{dx}\displaystyle\int_{0}^{x^{2}} \sin(t)\,dt\)?
  4. What does the first part of the fundamental theorem of calculus state?
  5. Evaluate the area under one arch of the sine: \(\displaystyle\int_0^{\pi}\sin x\,dx\).
  6. If \(F(x)=\displaystyle\int_1^x\cos(t^2)\,dt\), what is \(F'(x)\)?
  7. Let \(F(x)=\displaystyle\int_0^x (t^2-4)\,dt\). At which \(x>0\) does \(F\) have a local minimum?
  8. If \(F(x)=\displaystyle\int_x^{x^2}t\,dt\), what is \(F'(x)\)?
  9. Someone computes \(\displaystyle\int_{-1}^{1}\frac{dx}{x^2}=\Big[-\frac1x\Big]_{-1}^{1}=-2\). What is the error?

References

  • James Stewart, Calculus: Early Transcendentals, 8.ª edición, Cengage Learning, 2015.
  • Michael Spivak, Calculus, 4.ª edición, Publish or Perish, 2008.
  • Tom M. Apostol, Calculus, Vol. 1, 2.ª edición, John Wiley & Sons, 1967.
  • Stephen Abbott, Understanding Analysis, 2nd edition, Springer, 2015. §7.5, Teorema 7.5.1
  • Walter Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill, 1976. Cap. 6, Teorema 6.20
  • Robert G. Bartle, Donald R. Sherbert, Introduction to Real Analysis, 4th edition, Wiley, 2011. §7.3