2.5: Numerical Integration

Definitions

• Closed Form
• Numerical Integration
• Absolute Error
• Relative Error

Definition: Absolute Error

The absolute error, \( E_* \), of a numerical approximation, \( A \), to the true value of the quantity, \( T \), is defined to be\[ E_* = |T - A|. \nonumber \]The notation "\( * \)" is reserved for the method being used. For example, the absolute error if using the Midpoint Rule with \( n = 10 \) to approximate the value of a definite integral is\[ E_M = \left| T - M_{10} \right|. \nonumber \]

Definition: Relative Error

The relative error of an approximation is the error as a percentage of the actual value and is given by\[\left| \dfrac{T − A_*}{T}\right| \cdot 100\% = \dfrac{E_*}{|T|} \cdot 100\%. \nonumber \]

Theorems

• Midpoint Rule (derive)
• Trapezoidal Rule (derive)
• Simpson's Rule (derive)
• Error Bounds for Midpoint, Trapezoidal, and Simpson's Rules (do not derive)

Theorem: Midpoint Rule

Let \( f(x)\) be continuous on \([a,b]\), \( n\) be a positive integer, and \( \Delta x=\frac{b−a}{n}\). If \( [a,b]\) is divided into \( n\) subintervals, each of length \( \Delta x\), and \( m_i = \frac{x_{i - 1} + x_i}{2}\) is the midpoint of the \( i^{\text{th}}\) subinterval, set\[M_n = \sum_{i=1}^n f(m_i) \Delta x = \sum_{i=1}^n f\left(\dfrac{x_{i - 1} + x_i}{2}\right) \Delta x. \nonumber \]Then\[ \lim_{n \to \infty } M_n = \int ^b_af(x)\,dx.\nonumber \]

Theorem: The Trapezoidal Rule

Let \(f(x)\) be continuous over \([a,b]\), \(n\) be a positive integer, and \( \Delta x=\frac{b−a}{n}\). If \( [a,b]\) is divided into \(n\) subintervals, each of length \( \Delta x\), with endpoints at \( P=\{x_0,x_1,x_2, \ldots ,x_n\}\), set\[T_n=\dfrac{ \Delta x}{2}\left[f(x_0)+2 f(x_1) + 2 f(x_2) + \cdots + 2 f(x_{n−1}) + f(x_n)\right]. \nonumber \]Then,\[ \lim_{n \to \infty}T_n = \int ^b_af(x)\,dx. \nonumber \]

Theorem: Simpson's Rule

Assume that \(f(x)\) is continuous over \([a,b]\). Let \(n\) be a positive even integer and \( \Delta x=\frac{b−a}{n}\). Let \([a,b]\) be divided into \(n\) subintervals, each of length \( \Delta x\), with endpoints at \(P=\{x_0,x_1,x_2, \ldots ,x_n\}\). Set\[S_n = \dfrac{\Delta x}{3}\left[f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+2f(x_4) + \cdots +2f(x_{n−2})+4f(x_{n−1})+f(x_n)\right]. \nonumber \]Then,\[\lim_{n \to \infty }S_n= \int ^b_af(x)\,dx.\nonumber \]

Proof

WLOG, assume \( x_{i - 1} = -h \), \( x_i = 0 \), and \( x_{i + 1} = h \). The area under the curve is\[ \int_{-h}^{h} Ax^2 + Bx + C  \, dx = \dfrac{h}{3} \left( 2Ah^2 + 6C \right); \nonumber \]however, we also know that\[ \begin{array}{rcl}
y_0 & = & Ah^2 - Bh + C \\
y_1 & = & C \\
y_2 & = & Ah^2 + Bh + C \\
\end{array} \nonumber \]Thus,\[ y_0 + 4y_1 + y_2 = 2Ah^2 + 6C. \nonumber \]The rest requires adding up the parabolic sections.

Theorem: Error Bounds for the Midpoint and Trapezoidal Rules

Let \(f(x)\) be a continuous function over \([a,b]\), where \(f^{\prime\prime}(x)\) exists over this same interval interval. If \(M\) is the maximum value of \(\left|f^{\prime\prime}(x)\right|\) over \([a,b]\), then the upper bounds for the error in using \(M_n\) and \(T_n\) to estimate \(\displaystyle \int ^b_af(x)\, dx\) are

\[E_M \leq \dfrac{M(b−a)^3}{24n^2} \nonumber \]

and

\[E_T \leq \dfrac{M(b−a)^3}{12n^2}. \nonumber \]

Theorem: Error Bound for Simpson's Rule

Let \(f(x)\) be a continuous function over \([a,b]\) having a fourth derivative, \( f^{(4)}(x)\), over this interval. If \(M\) is the maximum value of \(\left|f^{(4)}(x)\right|\) over \([a,b]\), then the upper bound for the error in using \(S_n\) to estimate \(\displaystyle \int ^b_af(x)\, dx\) is given by\[E_S \leq \dfrac{M(b−a)^5}{180n^4}. \nonumber \]

Examples

Lecture Example \(\PageIndex{1}\)

Use \( M_4 \), \( T_4 \), and \( S_4 \) to approximate the value of\[ \int_{0}^{2} e^{x^2} \, dx. \nonumber \]If the true value of this integral is approximately \(16.45262776\), compute the absolute error and relative error in your approximations. Which method looks to be better?

Online Lecture Example \(\PageIndex{2}\)

Use \( M_{100} \) and \( T_{100} \) to approximate the value of\[ \int_{-2}^{7} \sin\left( \sqrt[3]{1 - x^2} e^{-x} \right) \, dx. \nonumber \]

Online Lecture Example \(\PageIndex{3}\)

How large do we have to take \( n \) so that the approximations of \( T_n \), \( M_n \), and \( S_n \) to the integral\[ \int_{0}^{\pi} \sin(x) \, dx \nonumber \]are accurate to within \( 0.00001 \)?

Online Lecture Example \(\PageIndex{4}\)

Approximate the surface area of the solid obtained by rotating the given function about the \( x \)-axis using \( S_{10} \).\[ y = x e^x, \quad 0 \leq x \leq 1. \nonumber \]