Simpson's 1/3rd Rule is a numerical integration method used to approximate the definite integral of a function. It is based on approximating the function by a quadratic polynomial within each subinterval.


The given function is ∫(0 to 6) sin(x) dx.


To apply Simpson's 1/3rd Rule, we need to divide the interval [0, 6] into subintervals of equal width. Since the width given is 1, we can divide the interval as follows:
- Subinterval 1: [0, 1]
- Subinterval 2: [1, 2]
- Subinterval 3: [2, 3]
- Subinterval 4: [3, 4]
- Subinterval 5: [4, 5]
- Subinterval 6: [5, 6]


To approximate the integral using Simpson's 1/3rd Rule, we use the formula:
∫(a to b) f(x) dx ≈ (h/3) [f(a) + 4f(a+h) + f(b)]

In this case, a = 0, b = 6, and h = 1 (width of each subinterval).

For each subinterval, we calculate the value of f(x) (sin(x)) at both ends and the midpoint, and substitute these values into the formula to get the approximation.

For the given function, the calculation for each subinterval is as follows:
- Subinterval 1: ∫(0 to 1) sin(x) dx ≈ (1/3) [sin(0) + 4sin(0.5) + sin(1)]
- Subinterval 2: ∫(1 to 2) sin(x) dx ≈ (1/3) [sin(1) + 4sin(1.5) + sin(2)]
- Subinterval 3: ∫(2 to 3) sin(x) dx ≈ (1/3) [sin(2) + 4sin(2.5) + sin(3)]
- Subinterval 4: ∫(3 to 4) sin(x) dx ≈ (1/3) [sin(3) + 4sin(3.5) + sin(4)]
- Subinterval 5: ∫(4 to 5) sin(x) dx ≈ (1/3) [sin(4) + 4sin(4.5) + sin(5)]
- Subinterval 6: ∫(5 to 6) sin(x) dx ≈ (1/3) [sin(5) + 4sin(5.5) + sin(6)]


To get the final approximation of the integral, we sum up the approximations for each subinterval:

Approximation ≈ ∫(0 to 1) sin(x) dx + ∫(1 to 2) sin(x) dx + ∫(2 to 3) sin(x) dx + ∫(3 to 4) sin(x

Divide the internal (0.6) into six parts each of width h = 1