Question

Use the Trapezoidal Rule with \(n = 10\) to approximate \(\int_0^{20} {\cos } (\pi x)dx\). Compare your result to the actual value. Can you explain the discrepancy?

Short answer

Using the trapezoidal rule the value of integral is \(20\) whereas with the actual method it is\(0\).

Step-by-step solution

Definition

Trapezoid Rule

If\((a,b)\)is divided into\(n\)subintervals, each of length\(\Delta x\), with endpoints at\(\left\{ {{x_0},{x_1},{x_2}, \ldots ,{x_n}} \right\}\), then\({T_n} = \frac{{\Delta x}}{2}\left( {f\left( {{x_0}} \right) + 2f\left( {{x_1}} \right) + 2f\left( {{x_2}} \right) + \cdots + 2f\left( {{x_{n - 2}}} \right) + 2f\left( {{x_{n - 1}}} \right) + f\left( {{x_n}} \right)} \right){\rm{ }}\)

Find length of interval

Consider the function\(f(x) = \cos (\pi x)\)

Limit of integration is from \(0\,to\,20\)so the interval is\( = (0,20)\)

And \({\rm{n}} = 10\)

Then \(\Delta x = \frac{{20}}{{10}} = 2\)

Then subintervals are \((0,2),(2,4),(4,6), \ldots \ldots (18,20)\).

By Trapezoidal rule

Using the trapezoidal rule mentioned above we have:

\(\begin{aligned}{c}\int_0^{20} {\cos } (\pi x)dx \approx {T_{10}}\\ = \frac{{\Delta x}}{2}(f(0) + 2(f(2) + f(4) + f(6) + \ldots . + f(18)) + f(20))\\ = 1 \cdot (\cos 0 + 2\{ \cos (2\pi ) + \cos (4\pi ) + - - + \cos (18\pi )\} + \cos (20\pi ))\end{aligned}\)

Since \(\cos 2n\pi = 1\)

Then \({T_{10}} = (1 + 2.(1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1) + 1)\)

\(\begin{aligned}{c}{T_{10}} = 1 + 18 + 1\\ = 20\end{aligned}\)

So, using trapezoidal rule the value of integral is\(\int_0^{20} {\cos } (\pi x)dx \approx 20\).

Actual value of integral

The value of actual integral is given by:

\(\begin{aligned}{c}\int_0^{20} {\cos } (\pi x)dx = \left( {\frac{{\sin (\pi x)}}{\pi }} \right)_0^{20}\,\,\,\,\,\,\,\,\,\,\,\,(As\,\int {\cos xdx = \sin x + C)} \\ = 0\end{aligned}\)

It is 0 because the actual integral gives the net area under the graph. Since the graph covers same area below the\(x\)-axis as above the\(x\) - axis so net area is\(\;0\).

But for \(n = 10\), in trapezoidal rule, the portion of the graph lying above\(x\)-axis is considered. So, this is representing the area lying only above\(x\)-axis.