Rectangle Sums from Rate Formulas

From a Rate Formula to a Rectangle Sum

A rate formula describes how quickly a quantity changes at each input value. To estimate the total change across an interval, divide the interval into smaller pieces, sample the rate once in each piece, and multiply each sampled rate by the piece's width. Adding these products gives a rectangle sum.

The main work in a formula-based problem is often careful bookkeeping: identify the subinterval width, calculate the appropriate sample locations, evaluate the formula at those locations, and include the width in the sum. The same procedure applies whether the samples are at left endpoints, right endpoints, or midpoints.

If \(r(t)\) measures a rate in units of quantity per unit of time, then \(\Delta t\) is measured in units of time. Each product \(r(t)\Delta t\) has units of quantity, as does the resulting sum. A rate formula can also take negative values; the corresponding rectangle contributes a negative product.

Sample Locations on an Equal Partition

Write the partition points as \(x_0,x_1,\ldots,x_n\), where \(x_0=a\) and \(x_n=b\). Because all subintervals have width \(\Delta t\), the partition points are \(x_i=a+i\Delta t\). The \(i\)th subinterval is \([x_{i-1},x_i]\).

Consequently, the left endpoint is \(a+(i-1)\Delta t\), the right endpoint is \(a+i\Delta t\), and the midpoint is halfway between those endpoints. Substituting these locations into \(r\) gives the sample values to use in the sums.

Here \(L_n\), \(R_n\), and \(M_n\) denote the left, right, and midpoint sums, respectively. In each formula, the index \(i\) runs from \(1\) through \(n\), so there is one sample for each subinterval. In particular, a left sum does not sample at \(x_n\), and a right sum does not sample at \(x_0\).

Why the Formulas Have This Form

Each subinterval contributes one rectangle. Its width is \(\Delta t\), and its height is the rate evaluated at the selected sample location. The rectangle's signed area is therefore the sampled rate multiplied by \(\Delta t\). Adding the \(n\) rectangle contributions gives the total estimate.

The sample-location formulas follow directly from the equal spacing. Starting at \(a\), moving \(i\) subinterval widths reaches \(x_i=a+i\Delta t\). The left endpoint of subinterval \(i\) is the preceding point \(x_{i-1}\). Its midpoint is the average of \(x_{i-1}\) and \(x_i\):

Thus the three formulas use the same interval widths and differ only in which point supplies each rectangle's height.

A Reliable Calculation Process

Worked Calculations

A Common Indexing Mistake

A midpoint is not the right endpoint of the subinterval. For example, on \([1,3]\) with \(n=2\), the width is \((3-1)/2=1\), and the subintervals are \([1,2]\) and \([2,3]\). Using the right endpoints \(2\) and \(3\) as if they were midpoints for \(r(t)=t^2+1\) gives

That is the right sum, not the midpoint sum. The actual midpoint locations are \(1.5\) and \(2.5\), found by taking the average of each pair of endpoints. Their rate values are \(r(1.5)=1.5^2+1=13/4\) and \(r(2.5)=2.5^2+1=29/4\). Thus the midpoint sum is

The correction is to calculate the requested locations before evaluating the formula. A location error changes which rate values are used, even if every substitution and addition afterward is correct.

What to Keep Track Of

For every rate-formula problem, distinguish three ingredients: the width, the sample locations, and the rate values. The width comes from the interval and the number of equal subintervals. The chosen sum determines the locations. The given formula determines the values at those locations.

Finally, multiply the rate values by the width before interpreting the result. A sum of rate values alone still has rate units; multiplying by time converts each rectangle contribution to units of accumulated quantity. These steps turn the formula into a definite, checkable finite calculation.