Use Simpson's Rule and all the data in the following table to estimate the value of the integral b=-5 a=-11
x= -11 -10 -9 -8 -7 -6 -5
y= 6 -9 -3 -9 -1 -9 -1
To estimate the value of the integral using Simpson's Rule, we need to divide the interval [a, b] into an even number of subintervals. Since the interval is from x = -11 to x = -5, we have a total of 7 data points. Therefore, we can divide the interval into three subintervals.
The formula for approximating the integral using Simpson's Rule is:
∫[a,b] y dx ≈ (h / 3) * [y0 + 4(y1) + 2(y2) + 4(y3) + 2(y4) + 4(y5) + y6]
Where:
- h is the step size (width of each subinterval), which is equal to (b - a) / n (where n is the number of subintervals)
- y0, y1, y2, ..., yn are the corresponding y-values of the data points.
Using the given data in the table, we have:
- a = -11
- b = -5
Step 1: Calculate the step size
h = (b - a) / n
= (-5 - (-11)) / 3
= 6 / 3
= 2
Step 2: Plug the y-values into the Simpson's Rule formula
∫[-11,-5] y dx ≈ (2 / 3) * [6 + 4(-9) + 2(-3) + 4(-9) + 2(-1) + 4(-9) + (-1)]
Simplifying the expression:
∫[-11,-5] y dx ≈ (2 / 3) * [-45 - 6 - 6 - 4 - 36 - 1]
= (2 / 3) * [-98]
Step 3: Calculate the final answer
∫[-11,-5] y dx ≈ (2 / 3) * (-98)
≈ -130.6667
Therefore, the estimated value of the integral ∫[-11,-5] y dx using Simpson's Rule is approximately -130.6667.