Use Simpson's Rule and all the data in the following table to estimate the value of the integral .
x -16 -15 -14 -13 -12 -11 -10
y -8 9 4 9 -5 -9 3
1/3[f(16)+4f(15)+2f(14)+4f(13)+2f(12)+4f(11)+f(10)]
I'm assuming for deltax is 1.
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To use Simpson's Rule, we need to have an even number of subintervals. Since we have 7 points, we can divide them into 3 subintervals: [-16, -14], [-14, -12], and [-12, -10].
The formula for Simpson's Rule for approximating an integral is:
∫f(x)dx ≈ (h/3) * [f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + 2f(x4) + ... + f(xn)]
Using the given data, we can calculate the integral as follows:
Subinterval 1: [-16, -14]
h = (-14 - (-16)) / 2 = 2
f(x0) = f(-16) = -8
f(x1) = f(-15) = 9
f(x2) = f(-14) = 4
Subinterval 2: [-14, -12]
h = (-12 - (-14)) / 2 = 2
f(x0) = f(-14) = 4
f(x1) = f(-13) = 9
f(x2) = f(-12) = -5
Subinterval 3: [-12, -10]
h = (-10 - (-12)) / 2 = 2
f(x0) = f(-12) = -5
f(x1) = f(-11) = -9
f(x2) = f(-10) = 3
Now we can substitute the values into the formula for Simpson's Rule:
∫f(x)dx ≈ (2/3) * [-8 + 4 + 4 + 9 + (-5) + (-5) + (-9) + 3]
Simplifying:
∫f(x)dx ≈ (2/3) * [-8 + 8 + 9 + (-5) + (-5) + (-9) + 3]
∫f(x)dx ≈ (2/3) * [-7]
Thus, the estimated value of the integral using Simpson's Rule for the given data is -14/3 or approximately -4.67.
To estimate the value of the integral using Simpson's Rule, we need to divide the interval from -16 to -10 into subintervals of equal width. In this case, the width of each subinterval is 1 since the difference between consecutive x-values is 1.
Since there are 7 x-values, we can divide them into 3 subintervals, which are -16 to -14, -14 to -12, and -12 to -10.
First, let's calculate the values of y for each x-value:
y(-16) = -8
y(-15) = 9
y(-14) = 4
y(-13) = 9
y(-12) = -5
y(-11) = -9
y(-10) = 3
Next, let's calculate the areas of the subintervals using Simpson's Rule:
Area of the first subinterval = (1/3) * (y(-16) + 4 * y(-15) + y(-14)) = (1/3) * (-8 + 4 * 9 + 4) = (1/3) * (36) = 12
Area of the second subinterval = (1/3) * (y(-14) + 4 * y(-13) + y(-12)) = (1/3) * (4 + 4 * 9 - 5) = (1/3) * (31) = 10.333...
Area of the third subinterval = (1/3) * (y(-12) + 4 * y(-11) + y(-10)) = (1/3) * (-5 + 4 * (-9) + 3) = (1/3) * (-41) = -13.666...
Finally, to estimate the value of the integral, we sum up the areas of all the subintervals:
Estimate = Area of the first subinterval + Area of the second subinterval + Area of the third subinterval
= 12 + 10.333... + (-13.666...)
= 8.666...
Therefore, the estimated value of the integral using Simpson's Rule and the given data is approximately 8.666.