use the midpoint rule to approximate the integral from -7 to 1 of (10x-5x^2)dx
To approximate the integral using the midpoint rule, you need to partition the interval [-7, 1] into equal subintervals and evaluate the midpoint of each subinterval within the integral expression.
Step 1: Determine the number of subintervals.
To apply the midpoint rule, you need to choose the number of subintervals, denoted by 'n'. A larger number of subintervals generally leads to a more accurate approximation. For this example, let's choose 'n' to be 4.
Step 2: Determine the width of each subinterval.
The width, 'Δx', of each subinterval can be computed by dividing the overall interval length by the number of subintervals.
Δx = (b - a) / n
Δx = (1 - (-7)) / 4
Δx = 8 / 4
Δx = 2
Step 3: Find the midpoint of each subinterval.
For each subinterval, find its midpoint by adding the lower endpoint of the subinterval to half of the subinterval width.
Midpoint of the i-th subinterval = (a + [(i - 1/2) * Δx])
where i = 1, 2, 3, ..., n.
For n = 4, the midpoints would be:
x1 = (-7 + [(1 - 1/2) * 2]) = -6
x2 = (-7 + [(2 - 1/2) * 2]) = -4
x3 = (-7 + [(3 - 1/2) * 2]) = -2
x4 = (-7 + [(4 - 1/2) * 2]) = 0
Step 4: Evaluate the function at each midpoint.
Evaluate the function, (10x - 5x^2), at each midpoint computed in the previous step.
f(x1) = (10 * (-6)) - (5 * (-6)^2) = -60 - 180 = -240
f(x2) = (10 * (-4)) - (5 * (-4)^2) = -40 - 80 = -120
f(x3) = (10 * (-2)) - (5 * (-2)^2) = -20 - 20 = -40
f(x4) = (10 * 0) - (5 * 0^2) = 0
Step 5: Approximate the integral using the midpoint rule.
The approximation of the integral using the midpoint rule can be found by multiplying the sum of the function evaluations at each midpoint by the subinterval width.
Approximation ≈ Δx * (f(x1) + f(x2) + f(x3) + f(x4))
≈ 2 * (-240 + (-120) + (-40) + 0)
≈ 2 * (-400)
≈ -800
Therefore, the midpoint rule approximation of the integral from -7 to 1 of (10x - 5x^2) dx is approximately -800.