The table gives the values of a function obtained from an experiment. Use them to estimate
9
f(x) dx
3
using three equal subintervals with right endpoints, left endpoints, and midpoints.
x 3 4 5 6 7 8 9
f(x) −3.5 −2.1 −0.5 0.2 0.8 1.3 1.8
(c) Estimate
9
f(x) dx
3
using three equal subintervals with midpoints.
M3 =?
To estimate the integral using three equal subintervals with midpoints, we can use the midpoint rule.
The midpoint rule states that the integral of a function over an interval can be approximated by the sum of the areas of rectangles, where the height of each rectangle is the value of the function at the midpoint of the corresponding subinterval.
To find the midpoint of each subinterval, we can take the average of the x-values of the endpoints. For example, the first subinterval would have a midpoint of (3+4)/2 = 3.5.
Let's calculate the estimate:
Subinterval 1:
Midpoint = (3+4)/2 = 3.5
Height = f(3.5) = -2.1 (interpolate between -3.5 and -2.1)
Area = 1*(4-3) = 1
Subinterval 2:
Midpoint = (4+5)/2 = 4.5
Height = f(4.5) = -0.5 (interpolate between -2.1 and -0.5)
Area = 0.5*(5-4) = 0.5
Subinterval 3:
Midpoint = (5+6)/2 = 5.5
Height = f(5.5) = 0.2 (interpolate between -0.5 and 0.2)
Area = 0.2*(6-5) = 0.2
Now, we can sum up the areas of all the subintervals:
M3 = Area1 + Area2 + Area3 = 1 + 0.5 + 0.2 = 1.7
Hence, the estimate for
9
f(x) dx
3
using three equal subintervals with midpoints is M3 = 1.7.