 # Definite integrals

## Fundamental Theorem of Calculus

If \$f\$ is a continuous function on the interval \$a ≤ x ≤ b\$ and \$F\$ is an antiderivative of \$f\$ on \$a ≤ x ≤ b\$, then:

\$\$∫↙{a}↖{b} f(x) dx = [F(x)]_a^b = F(b) - F(a)\$\$

### Definite Integral Properties

1. \$∫↙{a}↖{b} kf(x)dx = k∫↙{a}↖{b} f(x)dx\$
2. \$∫↙{a}↖{b} (f(x)±g(x))dx = ∫↙{a}↖{b} f(x)dx ± ∫↙{a}↖{b} g(x)dx\$
3. \$∫↙{a}↖{a} f(x)dx = 0\$
4. \$∫↙{a}↖{b} f(x)dx = -∫↙{b}↖{a} f(x)dx\$
5. \$∫↙{a}↖{b} f(x)dx = ∫↙{a}↖{c} f(x)dx + ∫↙{c}↖{b} f(x)dx\$
6. \$∫u⋅dv = uv - ∫v⋅du\$

### Definite Integral Properties

• Constant Rule: \$∫ kdx = kx + C\$
• Power Rule: \$∫x^n dx = 1/(n + 1) x^{n+1} + C\$, \$n ≠ 1\$
• Constant multiple rule: \$∫kf(x)dx = k ∫f(x) dx\$
• Sum of difference rule: \$∫(f(x) ± g(x))dx = ∫f(x)dx ± ∫g(x)dx\$
• \$e^x\$ and \$1/x\$ integrals: \$1/{x}dx = lnx + C\$, \$x > 0\$, \$∫e^xdx = e^x + C\$
• Linear composition: \$∫f(ax + b)dx = 1/{a}F(ax + b) + C\$, where \$F'(x) = f(x)\$

#### Solutions

Two common methods for solving integrals are:

Substitution: \${dy}/{dx} = {dy}/{du}⋅{du}/{dx}\$

Rearranging the equation to obtain the form \$∫f(g(x))g'(x)dx\$.

An indefinite integral is a family of functions that differ by the constant C. A definite integral is of the form:

\$\$∫↙a↖b f(x) dx\$\$

Leibniz invented the ∫ symbol for integration by the 1680s.

\$∫↙a↖b f(x) dx\$ is read as 'the integral from a to b of f(x) with respect to x'.

If a function \$f\$ is defined for a ≤ x ≤ b and \${lim}↙{n→∞}∑↙{i=1}↖n f(x_i)Δx_i\$ exists, then \$f\$ is integrable on a ≤ x ≤ b.

This is the definite integral and it is denoted as \$∫↙a↖b f(x) dx\$, where a and b are the lower and upper limits of integration.

If \$f\$ is a non-negative function for a ≤ x ≤ b, then \$∫↙a↖b f(x) dx\$ gives the area under the curve from x = a to x = b.

### Have you seen the C?

Definite integrals resolve to a number. This is because the constant C which is in the indefinite integral solution can be cancelled during the procedure of quantifying the definite integral. The definite integral from a to b of a function is the area between the curve from a to b and the x-axis.

From the graph of the function \$f(x) = 4 - x^2\$, determine the integral and use this to calculate the area under the curve from x = -2 to x = +2.

```\$A = ∫↙{-2}↖{2} f(x) dx = ∫↙{-2}↖{2} (4 - x^2) dx \$
\$= [4x - {x^3}/3 + C]_{-2}^{2} \$
\$= (8 - 8/3 + C) - (-8 + 8/3 + C) \$
\$= 16 - {16}/3 + C - C \$
\$= {32}/3 \$
\$= 10.67\$

```

## Area between two curves

If \$y_1\$ and \$y_2\$ are continuous on \$a ≤ x ≤ b\$ for all x in \$a ≤ x ≤ b\$, then the area between \$y_1\$ and \$y_2\$ from x = a to x = b is given by:
\$\$∫↙{a}↖{b} (y_1 - y_2)dx \$\$

### Area under the x-axis

The area of a curve is never negative. Therefore, the part of a function which graphs to below the x-axis needs to be calculated separately from the part above the x-axis, and made positive.

If \$f(x)\$ is negative between \$n\$ and \$p\$, the area between the curve of \$f(x)\$, the x-axis, and the lines \$x = n\$ and \$x = p\$ is:

\$\$|∫_n^pf(x)dx|\$\$

## Area between curves

If functions \$f\$ and \$g\$ are continuous in the interval [a, b], and \$f(x) ≥ g(x) for the region \$a ≤ x /le; b\$, the the area between the two graphs is:

\$\$A = ∫_a^bf(x)dx - ∫_a^bg(x)dx = ∫_a^b(f(x)- g(x)dx\$\$

Associated Mathematicians:

• Isaac Newton
• Gottfried Leibnitz
• Jakob Bernoulli
• Johann Bernoulli (I)

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