Estimate the area under the graph f(x)=3x^3 + 5 from x=-1 to x=5 using 12 mid points?

I got the right answer for using 6 mid points but what mid points would I use for 12?

To estimate the area under the graph using the midpoint rule with 12 midpoints, you need to divide the interval [x=-1, x=5] into 12 equal subintervals. Each subinterval will have its own midpoint, which you can use to evaluate the function and approximate the area.

To find the midpoints for 12 subintervals, you can use the formula:

xᵢ = a + (2i+1) * h / 2

where:
- xᵢ is the ith midpoint,
- a is the lower limit of integration (in this case, -1),
- h is the width of each subinterval (which can be found by dividing the total interval width by the number of subintervals), and
- i ranges from 0 to n-1, where n is the number of subintervals (in this case, 12).

Here's how you can find the midpoints for 12 subintervals:

1. Calculate the width of each subinterval:
Width of each subinterval (h) = (upper limit of integration - lower limit of integration) / number of subintervals
= (5 - (-1)) / 12
= 6 / 12
= 0.5

2. Calculate the midpoints:
For i = 0, 1, 2, ..., 11:
xᵢ = -1 + (2i+1) * (0.5) / 2
x₀ = -1 + (2 * 0 + 1) * (0.5) / 2 = -0.75
x₁ = -1 + (2 * 1 + 1) * (0.5) / 2 = -0.25
...
x₁₁ = -1 + (2 * 11 + 1) * (0.5) / 2 = 4.75

These values give you the midpoints for the 12 subintervals. You can now evaluate the function at each midpoint and use the midpoint rule formula to estimate the area under the graph.