Question

Use Simpson's $3/8^{th}$ rule, to estimate $\int_{1}^{7} f(x)dx$ from the following data.

x	1	2	3	4	5	6	7
f(x)	81	75	80	83	78	70	60

Answer 1

456

Answer 2

The problem requires us to estimate the integral $\int_{1}^{7} f(x)dx$ using Simpson's $3/8^{th}$ rule from the given data.

The formula for the composite Simpson's $3/8^{th}$ rule for $n$ intervals (where $n$ is a multiple of 3) is:

$\int_{a}^{b} f(x)dx \approx \frac{3h}{8} [y_0 + 3y_1 + 3y_2 + 2y_3 + 3y_4 + 3y_5 + y_6]$

From the data, we have:

$x_0 = 1, y_0 = f(1) = 81$
$x_1 = 2, y_1 = f(2) = 75$
$x_2 = 3, y_2 = f(3) = 80$
$x_3 = 4, y_3 = f(4) = 83$
$x_4 = 5, y_4 = f(5) = 78$
$x_5 = 6, y_5 = f(6) = 70$
$x_6 = 7, y_6 = f(7) = 60$

And the step size $h = 1$.

Substitute these values into the formula:

$\int_{1}^{7} f(x)dx \approx \frac{3(1)}{8} [81 + 3(75) + 3(80) + 2(83) + 3(78) + 3(70) + 60]$
$\int_{1}^{7} f(x)dx \approx \frac{3}{8} [81 + 225 + 240 + 166 + 234 + 210 + 60]$
$\int_{1}^{7} f(x)dx \approx \frac{3}{8} \times 1216 = 456$