Use the Midpoint Rule to approximate the integral (–10x+(7(x^2))) from 10 to 17 with n=3
To approximate the integral using the Midpoint Rule, follow these steps:
1. Determine the width of each subinterval. The width is given by (b - a) / n, where a and b are the limits of integration and n is the number of subintervals. In this case, a = 10, b = 17, and n = 3. Therefore, the width of each subinterval is (17 - 10) / 3 = 7/3.
2. Find the midpoint of each subinterval. The midpoint of each subinterval can be calculated using the formula (a + (i - 0.5) * width), where i represents the index of the subinterval (1, 2, 3, etc.) and width is the width of each subinterval. In this case, the midpoints are:
- For the first subinterval: 10 + (1 - 0.5) * (7/3) = 10.83
- For the second subinterval: 10 + (2 - 0.5) * (7/3) = 12.16
- For the third subinterval: 10 + (3 - 0.5) * (7/3) = 13.5
3. Evaluate the function at each midpoint. Substitute the midpoint value into the function (–10x + 7(x^2)) and calculate the corresponding function value. In this case, the function values are:
- For the first subinterval: –10(10.83) + 7(10.83^2) = 240.44
- For the second subinterval: –10(12.16) + 7(12.16^2) = 324.76
- For the third subinterval: –10(13.5) + 7(13.5^2) = 425.25
4. Calculate the approximate integral. Multiply the width of each subinterval by the corresponding function value at the midpoint, and then sum up these products. In this case, the approximate integral is:
(7/3) * (240.44 + 324.76 + 425.25) ≈ 336.22
Therefore, the approximate value of the integral (–10x + 7(x^2)) from 10 to 17 with n = 3 using the Midpoint Rule is approximately 336.22.