how do i approximate the area under a curce using simpson's rule?

Question

all i am given is a chart of x and y values
x .3 .6 .9 1.2 1.5 1.8 2.1
y 7.3 8.2 9.8 12.3 10.2 8.7 6.5
PLEASE HELP

Answer 1

For f(x) ≥ 0, the integral from a to b of f(x) dx equals the area under that part of the graph of f between (a,f(a)) and (b,f(b)) (and above the x-axis).

Simpson's Rule says that

integral from p to (p+2h) of f(x) dx ≈ (h/3)(f(p) + 4f(p+h) + f(p+2h))

So for your given data, the area under that part of the curve between (0.3, 7.3) and (0.9, 9.8) is approximately

((0.3)/3) (7.3 + 4*8.2 + 9.8)

where h=0.3 is the common difference between adjacent x-values.

Similarly, the area under that part of the curve between (0.9, 9.8) and (1.5, 10.2) is approximately

((0.3)/3) (9.8 + 4*12.3 + 10.2)

So if we combine these two estimates, the area under that part of the curve between (0.3, 7.3) and (1.5, 10.2) is approximately

((0.3)/3) (7.3 + 4*8.2 + 9.8 + 9.8 + 4*12.3 + 10.2)

which may be written

((0.3)/3) (7.3 + 4*8.2 + 2*9.8 + 4*12.3 + 10.2)

and you should be able to show that the area under the curve between (0.3, 7.3) and (2.1, 6.5) is approximately

((0.3)/3) (7.3 + 4*8.2 + 2*9.8 + 4*12.3 + 2*10.2 + 4*8.7 + 6.5)

This is an example of the so-called "composite" Simpson's Rule.

I'll let you do the arithmetic.