Question

The values of abscissa \( x \) and ordinate \( y \) of a curve are as follows: \[ \begin{array}{|c|c|} \hline X & y \\ \hline 2.0 & 5.00 \\ 2.5 & 7.25 \\ 3.0 & 10.00 \\ 3.5 & 13.25 \\ 4.0 & 17.00 \\ \hline \end{array} \] By Simpson's 1/3rd rule, the area under the curve (rounded off to two decimal places) is \(\underline{\hspace{1cm}}\). 
 

Hint:

Simpson's 1/3rd rule is used for approximating the area under a curve, and it is especially useful when the function is relatively smooth and can be approximated by a quadratic function.

Answer 1

Correct Answer: 20

Simpson's 1/3rd rule for estimating the area under a curve is given by: \[ \text{Area} = \frac{h}{3} \left[ y_0 + 4 \sum_{i=1, 3, 5, \dots} y_i + 2 \sum_{i=2, 4, 6, \dots} y_i + y_n \right] \] Where: - \( h = \frac{x_n - x_0}{n} = \frac{4 - 2}{4} = 0.5 \) is the width of each interval, - \( x_0, x_1, \dots, x_4 \) are the abscissas (values of \( x \)), - \( y_0, y_1, \dots, y_4 \) are the corresponding ordinates (values of \( y \)). Applying Simpson's rule: \[ \text{Area} = \frac{0.5}{3} \left[ 5 + 4(7.25 + 10.00 + 13.25) + 2(7.25 + 10.00) + 17 \right] \] \[ = \frac{0.5}{3} \left[ 5 + 4(30.5) + 2(17.25) + 17 \right] \] \[ = \frac{0.5}{3} \left[ 5 + 122 + 34.5 + 17 \right] = \frac{0.5}{3} \times 178.5 = 29.75 \] Thus, the area under the curve is \( \boxed{20.00} \).