# Numerical integration

### Numerical integration

Basic Concepts: Riemann sum

#### Lessons

Here are the three following ways to estimate the value of a definite integral with $n$ sub-intervals:

1) Midpoint Rule

$M_{n} = \int^b_a f(x)dx \approx \Delta x[f(x_{1})+f(x_{2})+...+f(x_{n-1})+f(x_{n})]$

Where $x_{i}$ is the midpoint of each interval.

2) Trapezoid Rule

$T_{n} = \int^b_a f(x)dx \approx \frac{\Delta x}{2} [f(x_{0})+2f(x_{1})+2f(x_{2})+...+2f(x_{n-1})+f(x_{n})]$

3) Simpsons Rule

$S_{n} = \int^b_a f(x)dx \approx \frac{\Delta x}{3} [f(x_{0})+4f(x_{1})+2f(x_{2})+...+2f(x_{n-2})+4f(x_{n-1})+f(x_{n})]$

If $f''$ is continuous from [$a, b$] and there is a value $M$ such that $|f''(x)| \leq M$ for all $x \in [a, b]$, then we can use the following formulas to calculate the error of the Midpoint and Trapezoid Rule:

1) Midpoint Rule Error Formula

$E_{M} \leq \frac{M(b-a)^{3}}{24n^{2}}$

2) Trapezoid Rule Error Formula

$E_{T} \leq \frac{M(b-a)^{3}}{12n^{2}}$

If $f^{(4)} (x)$ is continuous from $[a, b]$ and there is a value $K$ such that $|f^{(4)} (x)| \leq K$ for all $x \in [a, b]$, then we can use the following formulas to calculate the error of Simpsons Rule:

3) Simpson's Rule Error Formula

$E_{S} \leq \frac{K(b-a)^{5}}{180n^{4}}$

Here is a formula that may be of use when calculating the points of interest in Trapezoid and Simpsons Rule:

$x_{i} = a + i\Delta x$

Where $x_{i}$ is the point of interest at $i$.

• Introduction
Overview of Numerical Integration

- Midpoint, Trapezoid and Simpsons Rule

• 1.
Questions Regarding the Midpoint Rule

Approximate $\int^9_4 \sqrt{x} dx$ using Midpoint Rule with 5 sub-intervals.

• 2.
Approximate $\int^5_2 \frac{1}{2+x^{2}}$ using Midpoint Rule with 3 sub-intervals.

• 3.
Questions Regarding the Trapezoid Rule

Approximate $\int^1_0 e^{x} dx$ using Trapezoid Rule with 4 sub-intervals.

• 4.
Approximate $\int^5_1 x^{2} dx$ using Trapezoid Rule with 5 sub-intervals.

• 5.
Questions Regarding the Simpsons Rule

Approximate $\int^4_2 \sqrt{x-2} dx$ using Simpsons Rule with 4 sub-intervals.

• 6.
Approximate $\int^4_1 \ln (x^{2}) dx$ using Simpsons Rule with 6 sub-intervals.

• 7.
Questions Regarding Error Bounds

Let $f(x) = e^{x^{3}}$ consider $\int^1_0 e^{x^{3}} dx$. Assume you know that $|f''(x)| \leq 15e$ and $|f^{(4)}| \leq 585e$ for all $x \in [0, 1]$. If $n$ = 10, then find the following errors:

a)
$E_{M}$

b)
$E_{T}$

c)
$E_{S}$