Definite Integrals

A definite integral, written \int_a^b f(x)\,dx, is the signed area between the graph of f and the x-axis from x = a to x = b. It is defined as the limit of Riemann sums as the number of rectangles goes to infinity. Area above the x-axis counts as positive, and area below it counts as negative.

Before this lesson: Riemann Sums, Trapezoidal Rule and Circumference and Area of Circles

From Riemann sums to the definite integral

Riemann sums get closer to the true area as the rectangles get thinner. The definite integral is the exact value they approach. In the notation \int_a^b f(x)\,dx, the stretched S stands for “sum,” a and b are the limits of integration, f(x) is the integrand, and dx stands for the tiny width Δx.

\int_a^b f(x)\,dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i)\,\Delta x, \qquad \Delta x = \frac{b - a}{n}, \; x_i = a + i\,\Delta x

This limit exists for every continuous function on [a, b]. Any sample points (left, right or midpoints) give the same limit.

Signed area

When f(x) is negative, each rectangle’s “height” f(xᵢ) is negative, so its contribution is negative. That is why the definite integral is a net or signed area: the area above the axis minus the area below it.

Until you learn the Fundamental Theorem of Calculus, you can find many definite integrals exactly with geometry: rectangles, triangles, trapezoids and circles.

Definite integral examples

Example 1: Area of a trapezoid

Evaluate \int_0^4 (2x + 1)\,dx using geometry.

f(0) = 1, \; f(4) = 9

\frac{1 + 9}{2} \cdot 4 = 5 \cdot 4

\int_0^4 (2x + 1)\,dx = 20

Answer: 20.

Example 2: Area below the axis

Evaluate \int_{-2}^{3} x\,dx.

\frac{1}{2}(2)(2) = 2

\frac{1}{2}(3)(3) = 4.5

\int_{-2}^{3} x\,dx = 4.5 - 2 = 2.5

Answer: 2.5. The total shaded area is 6.5, but the integral is 2.5 because the part below the axis counts as negative.

Example 3: A semicircle

Evaluate \int_{-3}^{3} \sqrt{9 - x^2}\,dx.

y = \sqrt{9 - x^2} \Rightarrow x^2 + y^2 = 9, \; y \ge 0

\frac{1}{2}\pi(3)^2 = \frac{9\pi}{2}

\frac{9\pi}{2} \approx 14.14

Answer: \frac{9\pi}{2} \approx 14.14.

Example 4: The limit of Riemann sums

Evaluate \int_0^2 x^2\,dx as a limit of right Riemann sums.

\Delta x = \frac{2}{n}, \; x_i = \frac{2i}{n}

R_n = \sum_{i=1}^{n}\left(\frac{2i}{n}\right)^2 \cdot \frac{2}{n} = \frac{8}{n^3}\sum_{i=1}^{n} i^2

R_n = \frac{8}{n^3} \cdot \frac{n(n + 1)(2n + 1)}{6}

R_n = \frac{4(n + 1)(2n + 1)}{3n^2}

\lim_{n \to \infty}\frac{4(2n^2 + 3n + 1)}{3n^2} = \frac{8}{3}

Answer: \int_0^2 x^2\,dx = \frac{8}{3}, the value the Riemann sums 1.75, 2.75 and 3.75 were approaching.

Example 5: Word problem with displacement

A cart’s velocity is v(t) = 10 - 2t m/s for 0 \le t \le 8. Find its displacement and the total distance it travels.

v(t) = 0 \Rightarrow t = 5

\frac{1}{2}(5)(10) = 25

\frac{1}{2}(3)(6) = 9

\int_0^8 v(t)\,dt = 25 - 9 = 16

25 + 9 = 34

Answer: the displacement is 16 m (the cart ends 16 m forward), and the total distance traveled is 34 m. The graph above shows both regions.

Common mistakes

Related methods

Properties of definite integrals

Integrals over adjacent intervals add, constants factor out, and reversing the limits changes the sign. These rules come in the next lesson, Properties of Definite Integrals.

The fast way

Limits of sums are slow. The Fundamental Theorem of Calculus, later in this unit, evaluates definite integrals with antiderivatives: \int_0^2 x^2\,dx = \frac{2^3}{3} - 0 = \frac{8}{3} in one line.

Practice problems

1. Evaluate \int_0^3 4\,dx.
2. Evaluate \int_1^5 x\,dx.
3. Evaluate \int_0^6 (6 - x)\,dx.
4. Evaluate \int_{-4}^{0} x\,dx.
5. Evaluate \int_{-1}^{1} |x|\,dx.
6. Evaluate \int_0^2 \sqrt{4 - x^2}\,dx.
7. Evaluate \int_0^{2\pi} \sin x\,dx using symmetry.
8. Evaluate \int_0^3 (x - 1)\,dx.
9. Evaluate \int_0^1 x^3\,dx as a limit of right Riemann sums. Use \sum_{i=1}^{n} i^3 = \left(\frac{n(n + 1)}{2}\right)^2.
10. Water flows into a tank at r(t) = 2t liters per minute. Use a definite integral to find how much water flows in from t = 0 to t = 6.

\int_0^3 4\,dx = 3 \cdot 4 = 12

\frac{1 + 5}{2} \cdot 4 = 12

\frac{1}{2}(6)(6) = 18

-\frac{1}{2}(4)(4) = -8

\frac{1}{2}(1)(1) + \frac{1}{2}(1)(1) = 1

\frac{1}{4}\pi(2)^2 = \pi

\int_0^{\pi}\sin x\,dx = -\int_{\pi}^{2\pi}\sin x\,dx

-\frac{1}{2}(1)(1) + \frac{1}{2}(2)(2) = -0.5 + 2

R_n = \sum_{i=1}^{n}\left(\frac{i}{n}\right)^3\frac{1}{n} = \frac{1}{n^4}\left(\frac{n(n + 1)}{2}\right)^2

R_n = \frac{(n + 1)^2}{4n^2} \to \frac{1}{4}

\int_0^6 2t\,dt = \frac{1}{2}(6)(12) = 36

Definite integrals FAQ

What does a definite integral represent?

The signed area between the graph and the x-axis from a to b: area above the axis minus area below it. In context it is a total amount, such as distance from velocity.

What is the difference between a definite and an indefinite integral?

A definite integral has limits a and b and is a number. An indefinite integral has no limits and is a family of functions, F(x) + C.

Can a definite integral be negative?

Yes. If more of the area is below the x-axis than above it, the definite integral is negative.

How do you evaluate a definite integral without antiderivatives?

Use geometry when the graph makes simple shapes, or compute the limit of Riemann sums. The Fundamental Theorem of Calculus is the shortcut.

Summary

• \int_a^b f(x)\,dx = \lim_{n \to \infty}\sum f(x_i)\,\Delta x.
• It is a signed area: above the axis is positive, below is negative.
• Geometry gives exact values for lines and circles.
• Displacement is the integral of velocity; total distance adds the areas without signs.
• A definite integral is a number, with no + C.

Previous lesson: Trapezoidal Rule

Next lesson: Properties of Definite Integrals

Back to the Calculus 1 Index