The table gives values of a continuous function. Use the midpoint rule (n = 3) to estimate the average value of f on [10,40].

Question

x f(x)

10 19

15 34

20 48

25 30

30 55

35 62

40 46

Answer 1

The average value uses:

In this case a = 10 and b = 40 and f(x) is denoted by the table of values. To calculate the integral using the midpoint rule, break the table into sections 1) from 10 to 20 with x = 15 as the midpoint, 2) from 20 to 30 using 25 as the midpoint, and 3) from 30 to 40 using 35 as the midpoint. The integral will then be the sum of the areas with each area being its width (10) multiplied by the height at the midpoint or:

10•f(15) + 10•f(25) + 10•f(35)

So the average value is 1/(40 - 10)[10•f(15) + 10•f(25) + 10•f(35)] = (1/30)[10•34 + 10•30 + 10•62)

Answer 2

Step 1. Identify the goal of the problem

We are using the midpoint rule with three rectangles to estimate the area under the curve of the given function.

Step 2. Define the procedure

The midpoint rule divides the region into three equal width triangles and uses the function value at the midpoint of each domain to determine the height of the rectangles which will approximate the area under the curve.

Step 3. Compute

Width of domain: 40 - 10 = 30

Width of each rectangle: 30/3=10

Midpoint of each triangle: x = 15, 25, 35

f(x) at each midpoint: 34, 30, 62

Estimated Area under the graph is the sum of the three rectangles:

30*(30+34+62)=30*126=3780