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Justine's question at Yahoo! Answers regarding the Midpoint and Trapezoidal Rules
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Hello Justine,
First, let's look at the definitions of the two rules:
Midpoint Rule:
The Midpoint Rule is the approximation \(\displaystyle \int_a^b f(x)\,dx\approx M_n\), where
\(\displaystyle M_n=\frac{b-a}{n}\left[f\left(\frac{x_0+x_1}{2} \right)+f\left(\frac{x_1+x_2}{2} \right)+\cdots+f\left(\frac{x_{n-1}+x_{n}}{2} \right) \right]\)
Trapezoidal Rule:
The Trapezoidal Rule is the approximation \(\displaystyle \int_a^b f(x)\,dx\approx T_n\), where
\(\displaystyle T_n=\frac{b-a}{2n}\left[f\left(x_0 \right)+2f\left(x_1 \right)+\cdots+2f\left(x_{n-1} \right)+f\left(x_{n} \right) \right]\)
We can divide the interval $[a,b]$ into $2n$ equally spaced partitions, as so we may then write:
\(\displaystyle M_n=\frac{b-a}{n}\left[f\left(\frac{x_0+x_2}{2} \right)+f\left(\frac{x_2+x_4}{2} \right)+\cdots+f\left(\frac{x_{2n-2}+x_{2n}}{2} \right) \right]\)
And so using the midpoint formula, this becomes:
\(\displaystyle M_n=\frac{b-a}{2n}\left[2f\left(x_1 \right)+2f\left(x_3 \right)+\cdots+2f\left(x_{2n-1} \right) \right]\)
Likewise, the Trapezoidal Rule may now be written:
\(\displaystyle T_n=\frac{b-a}{2n}\left[f\left(x_0 \right)+2f\left(x_2 \right)+\cdots+2f\left(x_{2n-2} \right)+f\left(x_{2n} \right) \right]\)
Multiplying both by \(\displaystyle \frac{1}{2}\) and adding, we find:
\(\displaystyle \frac{1}{2}\left(T_n+M_n \right)=\frac{b-a}{2(2n)}\left[f\left(x_0 \right)+2f\left(x_1 \right)+\cdots+2f\left(x_{2n-1} \right)+f\left(x_{2n} \right) \right]\)
And using the definition of the Trapezoidal Rule, we find:
\(\displaystyle \frac{1}{2}\left(T_n+M_n \right)=T_{2n}\)
First, let's look at the definitions of the two rules:
Midpoint Rule:
The Midpoint Rule is the approximation \(\displaystyle \int_a^b f(x)\,dx\approx M_n\), where
\(\displaystyle M_n=\frac{b-a}{n}\left[f\left(\frac{x_0+x_1}{2} \right)+f\left(\frac{x_1+x_2}{2} \right)+\cdots+f\left(\frac{x_{n-1}+x_{n}}{2} \right) \right]\)
Trapezoidal Rule:
The Trapezoidal Rule is the approximation \(\displaystyle \int_a^b f(x)\,dx\approx T_n\), where
\(\displaystyle T_n=\frac{b-a}{2n}\left[f\left(x_0 \right)+2f\left(x_1 \right)+\cdots+2f\left(x_{n-1} \right)+f\left(x_{n} \right) \right]\)
We can divide the interval $[a,b]$ into $2n$ equally spaced partitions, as so we may then write:
\(\displaystyle M_n=\frac{b-a}{n}\left[f\left(\frac{x_0+x_2}{2} \right)+f\left(\frac{x_2+x_4}{2} \right)+\cdots+f\left(\frac{x_{2n-2}+x_{2n}}{2} \right) \right]\)
And so using the midpoint formula, this becomes:
\(\displaystyle M_n=\frac{b-a}{2n}\left[2f\left(x_1 \right)+2f\left(x_3 \right)+\cdots+2f\left(x_{2n-1} \right) \right]\)
Likewise, the Trapezoidal Rule may now be written:
\(\displaystyle T_n=\frac{b-a}{2n}\left[f\left(x_0 \right)+2f\left(x_2 \right)+\cdots+2f\left(x_{2n-2} \right)+f\left(x_{2n} \right) \right]\)
Multiplying both by \(\displaystyle \frac{1}{2}\) and adding, we find:
\(\displaystyle \frac{1}{2}\left(T_n+M_n \right)=\frac{b-a}{2(2n)}\left[f\left(x_0 \right)+2f\left(x_1 \right)+\cdots+2f\left(x_{2n-1} \right)+f\left(x_{2n} \right) \right]\)
And using the definition of the Trapezoidal Rule, we find:
\(\displaystyle \frac{1}{2}\left(T_n+M_n \right)=T_{2n}\)