- The definite integral of a positive function f(x) over an interval [a, b] is the area between f, the x-axis, x = a and x = b.
- The definite integral of a positive function f(x) from a to b is the area under the curve between a and b.
- If f(t) represents a positive rate (in y-units per t-units), then the definite integral of f from a to b is the total y-units that accumulate between t = a and t = b
Notation for the Definite Integral
The definite integral of
f from
a to
b is written
[latex] \int_a^b f(x)dx $
The [latex] \int $ symbol is called an
integral sign; it’s an elongated letter S, standing for sum. (The [latex] \int $ is actually the Σ from the Riemann sum, written in Roman letters instead of Greek letters.)
The
dx on the end must be included; you can think of [latex] \int $ and
dx as left and right parentheses. The
dx tells what the variable is—in this example, the variable is
x. (The
dx is actually the [latex] \Delta x $ from the Riemann sum, written in Roman letters instead of Greek letters.)
The function
f is called the
integrand.
The
a and
b are called the limits
of integration.
Verb Forms
We
integrate, or
find the definite integral of a function. This process is called
integration.
Formal Algebraic Definition
[latex] \int_{a}^{b} f(x)dx = {\lim_{n \to \infty}}_{\Delta x \to 0} \sum_{i = 1}^{n} f(x_i)\Delta x $
Practical Definition
The definite integral can be approximated with a Riemann sum (dividing the area into rectangles where the height of each rectangle comes from the function, computing the area of each rectangle, and adding them up). The more rectangles you use, the narrower the rectangles are, the better your approximation will be.
Looking Ahead
We will have methods for computing exact values of some definite integrals from formulas soon. In many cases, including when the function is given to you as a table or graph, you will still need to approximate the definite integral with rectangles.
The Definite Integral and Signed Area
- The definite integral of a function f(x) over an interval [a, b] is the signed area between f, the x-axis, x = a and x = b.
- The definite integral of a function f(x) from a to b is the signed area under the curve between a and b.
If the function is positive, the signed area is positive, as before (and we can call it area.)
If the function dips below the x-axis, the areas of the regions below the x-axis come in with a negative sign. In this case, we cannot call it simply “area.” These negative areas take away from the definite integral.
[latex] \int_{a}^{b} f(x)dx $ = (Area above x-axis) – (Area below x-axis).
If
f(
t) represents a positive rate (in
y-units per
t-units), then the
definite integral of
f from
a to
b is the
total y-units that accumulate between
t =
a and
t =
b.
If
f(
t) represents any rate (in
y-units per
t-units), then the
definite integral of
f from
a to
b is the
net y-units that accumulate between
t =
a and
t =
b.