Using 4 equal-width intervals, show that the trapezoidal rule is the average of the upper and lower sum estimates for the integral from 0 to 4 of x squared, dx.
recall the formula for the area of a trapezoid:
A = (B+b)h/2
Its area is the average of the areas of the rectangles formed using the lengths of its bases: the left and right function values!
To show that the trapezoidal rule is the average of the upper and lower sum estimates for the integral from 0 to 4 of x squared, dx using 4 equal-width intervals, we can follow these steps:
Step 1: Divide the interval [0, 4] into 4 equal-width subintervals. The width of each subinterval will be Δx = (4 - 0) / 4 = 1.
Step 2: Consider the lower sum estimate for the integral. The lower sum estimate is obtained by evaluating the function at the left endpoint of each subinterval and summing the results. In this case, we evaluate the function f(x) = x^2 at the left endpoints of the subintervals.
The left endpoints of the subintervals are 0, 1, 2, and 3. Evaluating f(x) = x^2 at these points gives:
f(0) = 0^2 = 0
f(1) = 1^2 = 1
f(2) = 2^2 = 4
f(3) = 3^2 = 9
Step 3: Calculate the lower sum estimate by summing the evaluations at the left endpoints:
Lower Sum Estimate = f(0) + f(1) + f(2) + f(3) = 0 + 1 + 4 + 9 = 14
Step 4: Similarly, consider the upper sum estimate for the integral. The upper sum estimate is obtained by evaluating the function at the right endpoint of each subinterval and summing the results. In this case, we evaluate f(x) = x^2 at the right endpoints of the subintervals.
The right endpoints of the subintervals are 1, 2, 3, and 4. Evaluating f(x) = x^2 at these points gives:
f(1) = 1^2 = 1
f(2) = 2^2 = 4
f(3) = 3^2 = 9
f(4) = 4^2 = 16
Step 5: Calculate the upper sum estimate by summing the evaluations at the right endpoints:
Upper Sum Estimate = f(1) + f(2) + f(3) + f(4) = 1 + 4 + 9 + 16 = 30
Step 6: Finally, calculate the trapezoidal rule estimate by averaging the lower and upper sum estimates:
Trapezoidal Rule Estimate = (Lower Sum Estimate + Upper Sum Estimate) / 2 = (14 + 30) / 2 = 22 / 2 = 11
Therefore, using the trapezoidal rule with 4 equal-width intervals, the estimate for the integral of x^2 from 0 to 4 is 11.
To demonstrate that the trapezoidal rule is the average of the upper and lower sum estimates for the given integral, we need to divide the interval [0, 4] into 4 equal-width intervals, calculate the upper and lower sum estimates, and then show that the trapezoidal rule is the average of those estimates.
1. Divide the interval [0, 4] into 4 equal-width subintervals:
The width of each subinterval can be found by dividing the total width of the interval (4 - 0 = 4) by the number of subintervals (4):
Width of subinterval = (4 - 0) / 4 = 1
The 4 equal-width subintervals are:
[0, 1], [1, 2], [2, 3], [3, 4]
2. Calculate the upper and lower sum estimates:
The upper sum estimate for each subinterval is the maximum value of the function within that subinterval multiplied by the width of the subinterval.
The lower sum estimate is the minimum value of the function within that subinterval multiplied by the width of the subinterval.
For the function f(x) = x^2, the maximum and minimum values within each subinterval are as follows:
[0, 1]: Maximum = 1^2 = 1, Minimum = 0^2 = 0
[1, 2]: Maximum = 2^2 = 4, Minimum = 1^2 = 1
[2, 3]: Maximum = 3^2 = 9, Minimum = 2^2 = 4
[3, 4]: Maximum = 4^2 = 16, Minimum = 3^2 = 9
Calculating the upper and lower sum estimates for each subinterval:
[0, 1]: Upper Sum Estimate = 1 * 1 = 1, Lower Sum Estimate = 0 * 1 = 0
[1, 2]: Upper Sum Estimate = 4 * 1 = 4, Lower Sum Estimate = 1 * 1 = 1
[2, 3]: Upper Sum Estimate = 9 * 1 = 9, Lower Sum Estimate = 4 * 1 = 4
[3, 4]: Upper Sum Estimate = 16 * 1 = 16, Lower Sum Estimate = 9 * 1 = 9
3. Apply the trapezoidal rule:
The trapezoidal rule calculates the average of the upper and lower sum estimates for each subinterval. This can be done by summing the upper and lower sum estimates for all subintervals and then dividing by the number of subintervals.
Sum of all Upper Sum Estimates = 1 + 4 + 9 + 16 = 30
Sum of all Lower Sum Estimates = 0 + 1 + 4 + 9 = 14
Applying the trapezoidal rule:
Trapezoidal Rule Estimate = (Sum of all Upper Sum Estimates + Sum of all Lower Sum Estimates) / (Number of Subintervals * 2)
= (30 + 14) / (4 * 2)
= 44 / 8
= 5.5
Therefore, the trapezoidal rule estimate for the given integral is 5.5, which is the average of the upper and lower sum estimates (30 and 14 respectively).