Use the Midpoint Rule to approximate the integral
from 2 to 7 S(–3x–7x^2)dx
with n=3.
See
http://www.jiskha.com/display.cgi?id=1244439041
A Riemann sum is an approximation to the area under a curve between defined limits by subdividing the area into n rectangles each of width h. The height is obtained by evaluating the function. If the height is evaluated at the left edge of each rectangle, it is called the left Riemann sum. Similar procedures apply to the Right and Middle Riemann sums.
See for example:
http://en.wikipedia.org/wiki/Riemann_sum
Notice that there is a typo in the equation for the left Riemann sum, the last term should be f(b-Q) where b is the upper limit, and Q is the width of the rectangles.
In your particular case,
a=2 (lower limit)
b=7 (upper limit)
n=3
h= (7-2)/3 = 5/3
f(x)=–3x–7x^2
middle Riemann sum
=(f(2+h/2)+f(2+3h/2)+f(2+5h/2))h
Post your results if you want a check.
To approximate the integral using the Midpoint Rule, we divide the interval [2, 7] into n subintervals of equal width. In this case, n=3, so we will have 3 subintervals.
Step 1: Determine the width of each subinterval.
The width, Δx, is given by (b - a) / n, where a and b are the endpoints of the interval. In this case, a = 2 and b = 7, so Δx = (7 - 2) / 3 = 5 / 3.
Step 2: Determine the midpoint of each subinterval.
The midpoint of a subinterval is given by (a + Δx/2) + iΔx, where i is the index of the subinterval (starting from 0). In this case, we have 3 subintervals (0, 1, 2), so the midpoints are:
- Subinterval 0: (2 + (5/3)/2) + 0*(5/3) = 2 + 5/6 = 17/6
- Subinterval 1: (2 + (5/3)/2) + 1*(5/3) = 2 + 5/6 + 5/3 = 27/6
- Subinterval 2: (2 + (5/3)/2) + 2*(5/3) = 2 + 5/6 + 2*(5/3) = 37/6
Step 3: Evaluate the function at each midpoint and sum the results.
Evaluate the function S(-3x - 7x^2) at each midpoint:
- S(17/6): S(-3*(17/6) - 7*(17/6)^2)
- S(27/6): S(-3*(27/6) - 7*(27/6)^2)
- S(37/6): S(-3*(37/6) - 7*(37/6)^2)
Finally, we can approximate the integral using the Midpoint Rule by multiplying the sum of the function evaluations at the midpoints by the width of each subinterval and summing these products:
Approximation = Δx * (S(17/6) + S(27/6) + S(37/6))
Remember to substitute S(x) with the actual function you are working with.