Use Simpson's Rule and all the data in the following table to estimate the value of the integral (with 7 at the bottom, 13 at the top) ydx.
x 7 8 9 10 11 12 13
y -5 -2 6 7 8 0 -4
To estimate the value of the integral using Simpson's Rule, we need to divide the interval from 7 to 13 into subintervals and calculate the area under the curve for each subinterval.
First, we need to determine the step size (h) by dividing the range (13 - 7) by the number of subintervals (n = 6):
h = (13 - 7) / 6 = 1
Next, we can calculate the area under the curve for each subinterval using Simpson's Rule formula:
∫ ydx ≈ [(h/3) * (y0 + 4y1 + 2y2 + 4y3 + 2y4 + 4y5 + y6)]
For the given data:
x 7 8 9 10 11 12 13
y -5 -2 6 7 8 0 -4
Plugging the values into the formula:
∫ ydx ≈ [(1/3) * (-5 + 4(-2) + 2(6) + 4(7) + 2(8) + 4(0) + (-4))]
Simplifying:
∫ ydx ≈ [(1/3) * (-5 - 8 + 12 + 28 + 16 + 0 - 4)]
∫ ydx ≈ [(1/3) * 39]
∫ ydx ≈ 13
Therefore, the estimated value of the integral ∫ ydx from x = 7 to x = 13 using Simpson's Rule is approximately 13.
To estimate the value of the integral using Simpson's Rule, we need to divide the range of integration into subintervals and obtain the values of y for each point. However, the given data does not provide enough points for each subinterval. We need at least two points per subinterval in order to use Simpson's Rule.
In this case, the given data consists of only one point per subinterval. Therefore, we cannot directly apply Simpson's Rule to estimate the integral.
To use Simpson's Rule, we need to have an odd number of subintervals. However, in this case, we have a single subinterval [7, 13], which does not fulfill this requirement.
If you have additional data points within the range 7 to 13, please provide them, and I will be happy to compute the integral estimation using Simpson's Rule.
There are lots of online calculators, such as
https://www.emathhelp.net/calculators/calculus-2/simpsons-rule-calculator/?f=1%2F%28x%5E5%2B7%29%5E%281%2F3%29&a=0&b=1&n=4&steps=on