Messed this up on my last question.
Given the table below for selected values of f(x), use 6 trapezoids to estimate the value of
∫f(x)dx where a=1 b=10
x 1 3 4 6 7 9 10
f(x) 4 8 6 10 10 12 16
Do u know ho to compare graphs?
why not show some work? You could use one of many good online trapezoidal rule calculators, as well.
My problem is the values I am looking for are in between the values given for f(x)...
Example, I am looking for f(2.5), f(5.5), f(8.5) all of which I am having to guess at from the values. What am I missing? Or is that the estimation part?
To estimate the value of ∫f(x)dx using 6 trapezoids, we can use the trapezoidal rule.
The trapezoidal rule approximates the area under a curve by dividing the region into trapezoids and summing up their areas. Each trapezoid has a base of width Δx and a height equal to the average of the function values at its endpoints.
First, let's calculate the width of each trapezoid, Δx. To do this, we need to find the interval between each pair of consecutive x-values.
We start with the given x-values: 1, 3, 4, 6, 7, 9, 10.
The intervals between the x-values are:
Δx1 = 3 - 1 = 2
Δx2 = 4 - 3 = 1
Δx3 = 6 - 4 = 2
Δx4 = 7 - 6 = 1
Δx5 = 9 - 7 = 2
Δx6 = 10 - 9 = 1
Now that we have the widths of the trapezoids, let's calculate the average function value at the endpoints of each trapezoid.
The function values at the endpoints are:
f(1) = 4
f(3) = 8
f(4) = 6
f(6) = 10
f(7) = 10
f(9) = 12
f(10) = 16
The average function values at the endpoints are:
Avg1 = (f(1) + f(3))/2 = (4 + 8)/2 = 6
Avg2 = (f(3) + f(4))/2 = (8 + 6)/2 = 7
Avg3 = (f(4) + f(6))/2 = (6 + 10)/2 = 8
Avg4 = (f(6) + f(7))/2 = (10 + 10)/2 = 10
Avg5 = (f(7) + f(9))/2 = (10 + 12)/2 = 11
Avg6 = (f(9) + f(10))/2 = (12 + 16)/2 = 14
Now we can calculate the area of each trapezoid.
Area1 = Δx1 * Avg1 = 2 * 6 = 12
Area2 = Δx2 * Avg2 = 1 * 7 = 7
Area3 = Δx3 * Avg3 = 2 * 8 = 16
Area4 = Δx4 * Avg4 = 1 * 10 = 10
Area5 = Δx5 * Avg5 = 2 * 11 = 22
Area6 = Δx6 * Avg6 = 1 * 14 = 14
Finally, we sum up the areas of all the trapezoids to get the estimated value of the integral:
∫f(x)dx ≈ Area1 + Area2 + Area3 + Area4 + Area5 + Area6
≈ 12 + 7 + 16 + 10 + 22 + 14
≈ 81
Therefore, the estimated value of the integral ∫f(x)dx, where a = 1 and b = 10, using 6 trapezoids is approximately 81.