The left, right, Trapezoidal, and Midpoint Rule...

The left, right, Trapezoidal, and Midpoint Rule approximations were used to estimate $\int_{0}^{2} f(x) d x,$ where $f$ is the function whose graph is shown. The estimates were 0.7811 , $0.8675,0.8632,$ and 0.9540 , and the same number of sub- intervals were used in each case. (a) Which rule produced which estimate? (b) Between which two approximations does the true value of $\int_{0}^{2} f(x) d x$ lie?

The left, right, Trapezoidal, and Midpoint Rule approximations were used to estimate $\int_{0}^{2} f(x) d x,$ where $f$ is the function whose graph is shown. The estimates were 0.7811 ,
$0.8675,0.8632,$ and 0.9540 , and the same number of sub-
intervals were used in each case.
(a) Which rule produced which estimate?
(b) Between which two approximations does the true value
of $\int_{0}^{2} f(x) d x$ lie?

Key Concepts

Left Riemann Sum

The Left Riemann Sum is a numerical integration method where the function’s value at the left endpoint of each subinterval is taken to approximate the area under the curve. This method tends to underestimate or overestimate the integral depending on whether the function is increasing or decreasing over the subintervals, and its accuracy depends on the smoothness and monotonicity of the function.

Right Riemann Sum

The Right Riemann Sum is a numerical integration technique that uses the function’s value at the right endpoint of each subinterval to compute the area under the curve. Similar to the left sum, this method may overestimate or underestimate the integral based on the function’s behavior over the interval, particularly if the function is monotonic.

Trapezoidal Rule

The Trapezoidal Rule approximates the area under a curve by dividing the total interval into subintervals and then replacing the curve on each subinterval with a straight line connecting the endpoints. This results in a series of trapezoids whose areas are summed to estimate the integral. It effectively averages the left and right Riemann sums, often yielding an approximation that is more accurate than either endpoint method for functions with moderate curvature.

Midpoint Rule

The Midpoint Rule is a method of numerical integration where the function’s value at the midpoint of each subinterval is used to estimate the area under the curve. This approach usually provides a more accurate approximation than using just the endpoints, especially when the function is relatively smooth, because it tends to capture the central tendency of the function over each subinterval.

Error Analysis in Numerical Integration

Error Analysis in Numerical Integration involves comparing the different approximations obtained by various rules (such as left, right, trapezoidal, and midpoint) to understand their bias in underestimating or overestimating the true value of the integral. By analyzing these errors, one can determine bounds within which the exact value of the integral lies and draw conclusions about the accuracy of each method based on the function’s behavior, such as its monotonicity and concavity.

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