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 2 of x squared, dx .
What does it mean by "upper and lower estimates?"
How do I find these?
Of the left and right sums, one will overestimate and the other will underestimate.
Since the area of a trapezoid (with vertical bases h1 and h2) is (h1+h2)*w
You can see how that is the average of the left-side height and the right-side height for each sub-interval.
I can't believe this is not explained in your text, and in far greater detail, giving illustrations!
To show that the trapezoidal rule is the average of the upper and lower sum estimates, let's first calculate the upper and lower sum estimates for the given integral.
The trapezoidal rule estimates the integral by dividing the interval into multiple sub-intervals and approximating the area under the curve within each sub-interval using trapezoids.
In this case, we are dividing the interval [0, 2] into 4 equal-width intervals. Therefore, the width of each sub-interval is (2 - 0) / 4 = 0.5.
Now, let's calculate the upper sum estimate:
1. Find the values of x at the endpoints of each sub-interval:
x₀ = 0
x₁ = 0.5
x₂ = 1
x₃ = 1.5
x₄ = 2
2. Calculate f(x) for each x:
f(x₀) = (x₀)² = 0² = 0
f(x₁) = (x₁)² = 0.5² = 0.25
f(x₂) = (x₂)² = 1² = 1
f(x₃) = (x₃)² = 1.5² = 2.25
f(x₄) = (x₄)² = 2² = 4
3. Calculate the areas of the trapezoids within each sub-interval:
A₁ = (0.5) * (f(x₀) + f(x₁)) = (0.5) * (0 + 0.25) = 0.125
A₂ = (0.5) * (f(x₁) + f(x₂)) = (0.5) * (0.25 + 1) = 0.625
A₃ = (0.5) * (f(x₂) + f(x₃)) = (0.5) * (1 + 2.25) = 1.625
A₄ = (0.5) * (f(x₃) + f(x₄)) = (0.5) * (2.25 + 4) = 3.125
4. Calculate the upper sum estimate:
Upper Sum Estimate = A₁ + A₂ + A₃ + A₄ = 0.125 + 0.625 + 1.625 + 3.125 = 5.5
Next, let's calculate the lower sum estimate:
1. Calculate the areas of the rectangles within each sub-interval:
R₁ = (0.5) * f(x₀) = (0.5) * 0 = 0
R₂ = (0.5) * f(x₁) = (0.5) * 0.25 = 0.125
R₃ = (0.5) * f(x₂) = (0.5) * 1 = 0.5
R₄ = (0.5) * f(x₃) = (0.5) * 2.25 = 1.125
2. Calculate the lower sum estimate:
Lower Sum Estimate = R₁ + R₂ + R₃ + R₄ = 0 + 0.125 + 0.5 + 1.125 = 1.75
Finally, let's calculate the average of the upper and lower sum estimates:
Average = (Upper Sum Estimate + Lower Sum Estimate) / 2
= (5.5 + 1.75) / 2
= 7.25 / 2
= 3.625
Therefore, the trapezoidal rule is the average of the upper and lower sum estimates for the given integral.