6 Integrals

These notes are reusing text and imagery from: Calculus Volume 1 from OpenStax, Print ISBN 193816802X, Digital ISBN 1947172131, https://www.openstax.org/details/calculus-volume-1

6.1 Antiderivatives

Definition: A function \(F\) is an antiderivative of the function \(f\) if \[F'(x) = f(x)\] for all \(x\) in the domain of \(f\).

For example, for \(f(x) = 2x\) we use the power rule of differentiation and conclude that its antiderivative is \(F(x) = x^2\) because \(F'(x) = 2x\). But any function \(G(x) = x^2 + C\), where \(C\) is a constant, has \(G'(x) = 2x\) and therefore is also an antiderivative.

The general form of antiderivative is thus \(F(x) + C\).

6.1.1 Indefinite integrals

Given a function \(f\) we indicate by \(f'\) or \(\frac{df}{dx}\) its derivative.

If \(F\) is an antiderivative of \(f\) we write \[\int f(x) dx = F(x) + C \enspace ,\] where \(\int\) is the integral sign and \(\int f(x) dx\) is called the indefinite integral of \(f\) with respect to \(x\).

The rules for evaluating integrals come directly from the rules for evaluating derivatives.

Integrals behave in many ways as sums. For example: \[ \text{sums and differences} \qquad \int \big( f(x) \pm g(x) \big) dx = \int f(x) dx \pm \int g(x) dx \\ \text{constant multiple} \qquad \int c f(x) dx = c \int f(x) dx \]

6.2 Definite integrals

How do you approximate the area under the curve \(f(x)\) over a closed interval \([a, b]\)?

Idea: split the interval \([a,b]\) into \(n\) regular partitions with separating points \(\{x_0, x_1, x_2, \ldots, x_n \}\) so that \(x_0 = a\) and \(x_n = b\). The width of each partition (the distance between any two consecutive points x) is \[\Delta x = x_i - x_{i-1} = \frac{b-a}{n} \enspace .\]

We can then use these partitions as a basis of our estimation.

  1. Over each partition \([x_{i-1}, x_i]\) draw a rectangle with height \(f(x_{i-1})\).
  2. Calculate the area of the rectangle \(R_{i} = f(x_{i-1}) \Delta x\).
  3. To approximate the area under the curve \(f(x)\), sum the areas of all the rectangles between \(a\) and \(b\) \[A \approx \sum_{i=1}^n R_{i} = \sum_{i=1}^n f(x_{i-1}) \Delta x = f(x_0) \Delta x + f(x_0) \Delta x + \cdots + f(x_{n-1}) \Delta x \enspace .\]

This approach is called the left-endpoint approximation because it uses the left points for the height of the rectangles.

The right-endpoint approximation uses the right points for the height of the rectangles. \[A \approx \sum_{i=1}^n R_{i} = \sum_{i=1}^n f(x_i) \Delta x = f(x_1) \Delta x + f(x_2) \Delta x + \cdots + f(x_n) \Delta x \enspace .\]

Left-endpoint (a) and right-endpoint (b) approximation of the area under the curve.

We can see that when using a small number of intervals \(n\) with large width, neither the left nor the right approximations are particularly good.

However, when we increase the number of intervals to make the rectangles slim, the approximation improves and the left- and right- get closer to each other.

6.2.1 Riemann sums

There is no strong reason to use the left- or right- endpoint to measure the height of the rectangle. We can use the height \(f(x_i^*)\) at any point \(x_i^* \in [x_{i-1}, x_i]\).

Definition: Let \(f\) be a function defined on closed interval \([a, b]\) which is partitioned into \(n\) regular intervals with width \(\Delta x = x_i - x_{i-1}\). For each interval the point \(x_i^*\) is an point \(x_i^* \in [x_{i-1}, x_i]\). The Riemann sum is defined for \(f(x)\) as \[\sum_{i=1}^n f(x_i^*) \Delta x \enspace .\]

Definition: Let \(f\) be a continuous non-negative function on interval \([a,b]\) and let \(\sum_{i=1}^n f(x_i^*) \Delta x\) be the Riemann sum. Then the area under the curve \(f(x)\) on \([a,b]\) is given by \[A = lim_{n \to \infty} f(x_i^*) \Delta x \enspace .\]

Riemann sums graph

6.2.2 Definite integral

Definition: Let \(f\) be function defined on interval \([a,b]\) and let \(\sum_{i=1}^n f(x_i^*) \Delta x\) be the Riemann sum. Then the definite integral of \(f\) from \(a\) to \(b\) is given by \[\int_a^b f(x) dx = lim_{n \to \infty} f(x_i^*) \Delta x \enspace ,\] provided the limit exists. If the limit exists, the function is called integrable on \([a, b]\).

If a function \(f\) is continuous on \([a,b]\), then \(f\) is integrable \([a,b]\). (But some discontinuous functions are also integrable.)

We call the values \(a\) and \(b\) the limits of integration, \(x\) is the variable of integration and \(f(x)\) the integrand. Note that \(x\) has no effect on the computation and the integral will evaluate to the same number \[\int_a^b f(x) dx = \int_a^b f(v) dv = \int_a^b f(u) du \enspace .\]

Careful: The definite integral \(\int_a^b f(x) dx\) will give you a single number (and you can think of it as area under the curve between two points). The indefinite integral \(\int f(x) dx\) represents the antiderivative and therefore a family of functions in the form \(F(x) + C\).

If a function is not non-negative the definite integral can be thought of as the net signed area.

For example for the function in the image below we have \[\int_0^2 f(x) dx = lim_{n \to \infty} f(x_i^*) \Delta x = A1 - A2 \enspace .\]

6.2.2.1 Properties of definite integrals

Recall the sigma notation for sums

\[\sum_{i = 1}^n a_i = a_1 + a_2 + \cdots + a_n\] and its properties \[\sum_{i = 1}^n c = c + c \cdots + c = nc \\ \sum_{i = 1}^n c a_i = c a_1 + c a_2 + \cdots + c a_n = c \sum_{i = 1}^n a_i \\ \sum_{i = 1}^n ( a_i \pm b_i) = a_1 \pm b_1 + a_2 \pm b_2 + \cdots + a_n \pm b_n = \sum_{i = 1}^n a_i \pm \sum_{i = 1}^n b_i \\ \sum_{i = 1}^n a_i = a_1 + \cdots + a_m + a_{m + 1} + \cdots + a_n = \sum_{i = 1}^m a_i + \sum_{i = m+1}^n a_i \]

The properties of definite integrals closely resemble the properties of the sigma notation for sums \[\int_a^a f(x) dx = 0 \\ \int_a^b f(x) dx = - \int_b^a f(x) dx \\ \int_a^b \big( f(x) \pm g(x) \big) dx = \int_a^b f(x) dx \pm \int_a^b g(x) dx \\ \int_a^b c f(x) dx = c \int_a^b f(x) dx \\ \int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx \enspace . \]

6.3 Fundamental theorem of calculus

The fundamental theorem of calculus, part 1: If \(f\) is continuous on \([a, b]\) and the function \(F(x)\) is defined by \[F(x) = \int_a^x f(t) dt \enspace ,\] then \(F'(x) = f(x)\) over \([a, b]\) (\(F\) is antiderivative of \(f\) on \([a,b]\)).

The fundamental theorem of calculus, part 2 (evaluation): If \(f\) is continuous on \([a, b]\) and \(F\) is any antiderivative of \(f\), then \[\int_a^b f(x) dx = F(x) \big|_a^b = F(b) - F(a) \enspace .\]