Question

How do I integrate the curve numerically using the trapezoidal rule and integrate the curve numerically using Simposon's rule.

Provided is the data of absorbance values (y-axis) for each cuvette collection (x-axis). I also plotted the points on a graph. Please explain with all steps. Thanks.

56 DO YOU 0.035 0.048 0.075 0.116 0.156 0.201 0.247 0.29 0.327 0.352 0.369 0.373 0.367 0.345 0.324 0.284 0.255 0.219 0.189 0.161 0.129 0.11 0.093 0.077 0.065 0.058 0.049 0.043 0.04 0.038 18 30

Answer 1

Trapezoidal and Simpson rules are methods to approximately calculate a definite integral

Suppose there is a definite integral :

$\int_{a}^{b} f(x)dx$

where f(x) is the function, b is the last and a is the first limit of interval [a,b] over which we have to integrate this function.

In trapezoidal rule, we divide the graph into N equally spaced intervals. Like in our case we can divide it into 29 equally spaced intervals. For example, first interval from x = 1 to x = 2, second one from x = 2 to x = 3 and so on.

So by trapezoidal rule : -

$\int_{a}^{b} f(x)dx = \frac{\Delta x}{2} \left ( f(x_0) +f(x_n) + 2 \sum_{k = 1}^{n-1} f(x_k) \right )$

$\int_{a}^{b} f(x)dx = \frac{\Delta x}{2} \left ( f(x_0) + 2f(x_1) + 2f(x_2) +.... +2f(x_{n-1}) +f(x_n) \right ) \hspace{2cm} eqn(1)$

where $\Delta x = \frac{b-a}{N}$

and f(x₀) , f(x₁) , f(x₂) ... f(x_n) are the values of function at x = a , x = a + $\Delta x$ , x = a + 2$\Delta x$ ....x = a + N$\Delta x$ = b

Thus we have 29 intervals from x = 1 to x = 30

$\therefore \Delta x = \frac{30-1}{29} = 1$

At x = a = 1 , f(x₀) = 0.035

At x = a + $\Delta x$ = 2 , f(x₁) = 0.048

.......

At x = a + N$\Delta x$ = b = 30 , f(x_n) = 0.038

Just put all the values in the above equation (1)

Simpson rule : -

$\int_{a}^{b} f(x)dx = \frac{\Delta x}{3} \left ( f(x_0) + 4f(x_1) + 2f(x_2) +4f(x_3)+2f(x_4).... +4f(x_{n-1}) +f(x_n) \right ) \hspace{2cm} eqn(2)$

Where all the terms have the same meaning as discussed in trapezoidal rule. Simpson rule is a little bit different formula-wise.

NOTE : - Simpson rule only works when, number of intervals (N) in which the curve is divided is EVEN. Thus you can attempt this problem by using Simpson from x = 1 to x = 29 making 28 intervals (EVEN), and then applying trapezoidal from x = 29 to x = 30.