Question

using simpsons rule find 10to0 x^2 dx by taking n=10​

Answer 1

We need to find the value of $\large{\mathsf{\int_{10}^{0} x^{2}dx}}$ using Simpson's One-third Rule.

Working Formula -

$\large{\mathsf{I_{S}^{C}=\int_{x_{0}}^{x_{0}+nh}ydx}}$

  = $\large{\mathsf{\frac{h}{3}[y_{0}+y_{n}}}$

$\large{\mathsf{+4(y_{1}+y_{3}+...+y_{n-3}+y_{n-1})}}$

$\large{\mathsf{+2(y_{2}+y_{4}+...+y_{n-4}+y_{n-2})]}}$

{ see the attached table for the computational table }

$\large{\mathsf{h=\frac{x_{10}-x_{0}}{n}}}$

  $\large{\mathsf{=\frac{0-10}{10}}}$

  $\large{\mathsf{=\frac{-10}{10}}}$

  = - 1

From the table, we can find

$\large{\mathsf{I_{S}^{C} = \frac{h}{3}*(Total)}}$

    = $\large{\mathsf{\frac{-1}{3}*1000}}$

    = $\large{\mathsf{-\frac{1000}{3}}}$ ,

which is the required integral value, computed using Simpson's One-third Rule.

To confirm this value, let us try solving the sum using integration formulas.

Now, $\large{\mathsf{\int_{10}^{0} x^{2}dx}}$

      = $\large{\mathsf{[\frac{x^{3}}{3}]_{10}^{0}}}$

      = $\large{\mathsf{\frac{0}{3}-\frac{1000}{3}}}$

      = $\large{\mathsf{-\frac{1000}{3}}}$

We see that, the value of the integration in both the rules are same. Hence, verified.

Thus, solved.