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Calculus I Foundations (MATH 151) · Module 7 of 11

The Definite Integral & the Fundamental Theorem of Calculus

How summing infinitely many rectangles connects to antiderivatives.

In this module
  1. Riemann Sums & the Definite Integral
  2. Properties of Definite Integrals
  3. The Fundamental Theorem of Calculus, Part 1
  4. The Fundamental Theorem of Calculus, Part 2
  5. Check yourself

Riemann Sums & the Definite Integral

  • A Riemann sum approximates the area under a curve by summing the areas of rectangles; as the number of rectangles approaches infinity and their width approaches 0, the sum approaches the exact area, defining the definite integral ∫[a,b] f(x)dx.
A left Riemann sum
xyf(x) = x²
Four left rectangles of width 0.5 under f(x) = x² on [0, 2] give 0.5(0 + 0.25 + 1 + 2.25) = 1.75, an underestimate of the exact area 8/3 ≈ 2.67 because f is increasing.
Worked example

What does a Riemann sum approximate, and how does it relate to the definite integral?

Solution. It approximates the area under a curve using a finite number of rectangles; the definite integral is the exact value this approximation approaches as the number of rectangles goes to infinity and their width goes to zero.

Properties of Definite Integrals

  • ∫[a,b] f(x)dx = −∫[b,a] f(x)dx, so reversing the limits negates the integral; ∫[a,a] f(x)dx = 0; integrals are additive over adjacent intervals: ∫[a,b] + ∫[b,c] = ∫[a,c].
Worked example

If ∫[1,4] f(x)dx = 10, what is ∫[4,1] f(x)dx?

Solution. −10, since reversing the limits of integration negates the value of the integral.

The Fundamental Theorem of Calculus, Part 1

  • Part 1 states that if F(x) = ∫[a,x] f(t)dt, then F'(x) = f(x) — the derivative of an integral with a variable upper limit gives back the original function.
Worked example

If F(x) = ∫[0,x] (t²+1)dt, what is F'(x)?

Solution. By the Fundamental Theorem of Calculus Part 1, F'(x) = x² + 1 — the integrand itself, evaluated at x.

The Fundamental Theorem of Calculus, Part 2

  • Part 2 states that if F is any antiderivative of f, then ∫[a,b] f(x)dx = F(b) − F(a), letting definite integrals be evaluated using antiderivatives rather than limits of Riemann sums.
Worked example

Evaluate ∫[0,2] x² dx using the Fundamental Theorem of Calculus, Part 2.

Solution. An antiderivative of x² is F(x) = x³/3. The integral equals F(2) − F(0) = 8/3 − 0 = 8/3.

Check yourself

2 of this module's 5 quiz questions. Pick your answers, then check them in the app, with an explanation for every question.

1. What does a Riemann sum approximate?

2. What happens to a definite integral when you reverse its limits of integration?

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