A-level Mathematics/OCR/FP2/Complex Integration

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Midpoint Rule[edit]

The Midpoint Rule is more accurate than the Trapezium Rule. It works by finding the mid-points of rectangles drawn to the curve. The Midpoint Rule is:

\int_a^b f \left (x \right ) \,dx \approx = h \left [ f \left (x_1 \right ) + f \left (x_2 \right ) + \ldots + f \left (x_n \right )\right ]

Where: h = \frac{b-a}{n} n is the number of strips.

and x_i = \frac{1}{2} \left [ \left( a +\left \{i - 1 \right \} h \right) + \left (a + ih \right) \right]

Example[edit]

Use the Midpoint Rule to evaluate \int_{1}^{5}x^2 + 2x\ dx using 4 strips.

Firstly, we work out h.

h = \frac{5-1}{4} = \frac{1}{1} = 1

Now we begin to set up the Midpoint Rule.

\int_{1}^{5}x^2 + 2x\ dx \approx 1 \left [ f \left (x_1 \right ) + f \left (x_2 \right ) +  f \left (x_3 \right ) +  f \left (x_4 \right )\right ]

x_1 = \frac{1}{2} \left [ \left( 1 +\left \{1 - 1 \right \} 1 \right) + \left (1 + 1 \times 1 \right) \right] = \frac{1}{2} \left [ \left( 1 \right) + \left (2 \right) \right] = 1.5

x_2 = \frac{1}{2} \left [ \left( 1 +\left \{2 - 1 \right \} 1 \right) + \left (1 + 2 \times 1 \right) \right] = \frac{1}{2} \left [ \left( 2 \right) + \left (3 \right) \right] = 2.5

x_3 = \frac{1}{2} \left [ \left( 1 +\left \{3 - 1 \right \} 1 \right) + \left (1 + 3 \times 1 \right) \right] = \frac{1}{2} \left [ \left( 3 \right) + \left (4 \right) \right] = 3.5

x_4 = \frac{1}{2} \left [ \left( 1 +\left \{4 - 1 \right \} 1 \right) + \left (1 + 4 \times 1 \right) \right] = \frac{1}{2} \left [ \left( 4 \right) + \left (5 \right) \right] = 4.5

\int_{1}^{5}x^2 + 2x\ dx \approx 1 \left [ f \left (1.5 \right ) + f \left (2.5 \right ) +  f \left (3.5 \right ) +  f \left (4.5 \right )\right ]

Now we need to solve f(n)

\int_{1}^{5}x^2 + 2x\ dx \approx 1 \left [ 5.25 + 11.3 +  19.3 +  29.3\right ]

\int_{1}^{5}x^2 + 2x\ dx \approx 65

As you can see the resultant from the midpoint rule is closer to the true value 65\frac{1}{3} than the trapezium rule, but worse than Simpson's Rule.