Riemann Sums: Rules, Integral Limits, Error Bounds, and Higher Dimensions

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What is a Riemann sum?

A finite sum that approximates an integral by adding products of sampled function values and subinterval widths.

What is a partition?

An ordered set of points $a=x_0<x_1<\cdots<x_n=b$ that splits an interval into subintervals.

What does $\Delta x_i$ mean?

It is the width of the $i$th subinterval: $\Delta x_i=x_i-x_{i-1}$.

What is a sample point $x_i^*$?

A point chosen inside the $i$th subinterval; its function value is used as the height contribution for that piece.

Why refine a partition?

As the mesh shrinks, sums for an integrable function approach its integral.

Quiz questions

1. What is the main purpose of a Riemann sum?Answer: To estimate an accumulated quantity by summing contributions from partitioned pieces.
2. For an increasing, nonnegative function, how do left- and right-endpoint sums compare with the integral?Answer: The left sum underestimates, and the right sum overestimates.
3. In a three-dimensional Riemann sum, what does each cell-size factor represent?Answer: The volume of the corresponding cell.
4. For a continuous function on a closed interval, what happens to Riemann sums as the partition mesh tends to zero?Answer: They converge to the same definite integral for any sample-point choices.
5. Under the stated derivative bounds, how do midpoint and trapezoidal error bounds compare in order with left and right sums?Answer: They decrease on the order of $1/n^2$, rather than $1/n$.