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
it's clearly true, since the area of a trapezoid is the average of the two surrounding rectangles.
A = (b1 + b2)/2 * h
Now just do the tedious math to show it in this case
To show that the trapezoidal rule is the average of the upper and lower sum estimates for the integral, we first need to understand how the trapezoidal rule works.
The trapezoidal rule is a numerical method for approximating the value of a definite integral. It divides the interval of integration into subintervals and approximates the area under the curve within each subinterval by a trapezoid. The sum of the areas of these trapezoids gives an estimate of the integral.
Now, let's consider using 4 equal-width intervals to approximate the integral from 0 to 4 of x squared, dx using the trapezoidal rule. We divide the interval from 0 to 4 into four subintervals with width Δx = (4 - 0) / 4 = 1.
The formula for the trapezoidal rule is given by:
∫f(x) dx ≈ Δx/2 * (f(a) + 2f(a + Δx) + 2f(a + 2Δx) + 2f(a + 3Δx) + f(b))
where:
- f(x) is the function to be integrated (in this case, x^2).
- a and b are the lower and upper limits of integration (0 and 4 in this case).
Let's calculate the upper and lower sum estimates using the trapezoidal rule for the given function and interval:
Lower sum estimate:
Using the above formula:
∫x^2 dx ≈ Δx/2 * (f(a) + 2f(a + Δx) + 2f(a + 2Δx) + 2f(a + 3Δx) + f(b))
Substituting the values:
∫x^2 dx ≈ 1/2 * (f(0) + 2f(1) + 2f(2) + 2f(3) + f(4))
∫x^2 dx ≈ 1/2 * (0^2 + 2(1^2) + 2(2^2) + 2(3^2) + 4^2)
∫x^2 dx ≈ 1/2 * (0 + 2 + 8 + 18 + 16)
∫x^2 dx ≈ 1/2 * (44)
∫x^2 dx ≈ 22
Upper sum estimate:
Using the same formula from above:
∫x^2 dx ≈ Δx/2 * (f(a) + 2f(a + Δx) + 2f(a + 2Δx) + 2f(a + 3Δx) + f(b))
Substituting the values:
∫x^2 dx ≈ 1/2 * (f(0) + 2f(1) + 2f(2) + 2f(3) + f(4))
∫x^2 dx ≈ 1/2 * (0^2 + 2(1^2) + 2(2^2) + 2(3^2) + 4^2)
∫x^2 dx ≈ 1/2 * (0 + 2 + 8 + 18 + 16)
∫x^2 dx ≈ 1/2 * (44)
∫x^2 dx ≈ 22
We can see that both the upper and lower sum estimates for the integral from 0 to 4 of x squared, dx using 4 equal-width intervals using the trapezoidal rule are equal to 22.
Therefore, the trapezoidal rule provides the average of the upper and lower sum estimates for the integral.