From a Graph to an Accumulation Estimate
A graph can show how a quantity changes across an interval. If the vertical value represents a rate, the accumulated change over a time interval can be estimated by adding the contributions from a collection of rectangles. The graph supplies the heights; the interval supplies the widths.
This is useful when values are presented visually rather than in a table or formula. The basic task is to read the graph at selected locations, use those readings as rectangle heights, and add the rectangles’ signed areas. The estimate depends on the chosen partition and on whether the heights are read at left endpoints, right endpoints, or midpoints.
A graph-based answer is usually approximate: the plotted curve and its grid may not allow every height to be read exactly. The calculation should therefore preserve sensible precision and make clear which samples were used.
Reading the Graph and Setting Up the Partition
Suppose a graph represents a function \(f\) on an interval \([a,b]\). Partition the interval at points \(x_0,x_1,\ldots,x_n\), where \(x_0=a\), \(x_n=b\), and the points increase from left to right. The \(i\)th subinterval has width \(\Delta x_i=x_i-x_{i-1}\). Choose a sample location \(c_i\) in each subinterval and read the graph’s height there, \(f(c_i)\).
The corresponding rectangle has width \(\Delta x_i\) and signed height \(f(c_i)\). Its signed area is \(f(c_i)\Delta x_i\). Adding those contributions gives the rectangle sum.
For a left-endpoint sum, use \(c_i=x_{i-1}\). For a right-endpoint sum, use \(c_i=x_i\). For a midpoint sum, use \(c_i=(x_{i-1}+x_i)/2\). If the graph gives equal-width subintervals, each width is the same; otherwise, use each subinterval’s own width.
Read both axes before calculating. The horizontal axis tells you the \(x\)-locations and therefore the widths. The vertical axis gives the function values and therefore the heights. A vertical grid step of 2 means that moving one grid line vertically changes the height by 2, not by 1. If a sample lies between marked grid lines, estimate its height using the graph’s scale and record that it is an estimate.
Why the Rectangle Sum Has This Form
The method follows directly from the geometry of each rectangle. On one subinterval, the rectangle’s horizontal length is \(\Delta x_i\), and its vertical height is the sampled value \(f(c_i)\). The signed area of that rectangle is the product of these quantities. The total estimate is the sum of the signed areas across the partition.
Adding over all subintervals gives the displayed sum. This calculation estimates accumulated change because each rectangle treats the graph’s value as if it remained constant throughout its subinterval. A graph that varies within a subinterval is therefore represented by a constant-height approximation there. This explains both the method and why the result is an estimate rather than, in general, an exact total.
Worked Example: Left- and Right-Endpoint Sums
Worked Example: Left- and Right-Endpoint Sums
A graph on \([0,4]\) passes through the plotted points \((0,0)\), \((1,2)\), \((2,4)\), \((3,3)\), and \((4,1)\). Use four subintervals of width 1 to calculate the left- and right-endpoint rectangle sums.
The partition points are \(x_0=0\), \(x_1=1\), \(x_2=2\), \(x_3=3\), and \(x_4=4\). Each width is \(1\). The graph heights at these points, in order, are \(0,2,4,3,1\).
For the left sum, use the first endpoint of each subinterval. The heights are \(f(0)=0\), \(f(1)=2\), \(f(2)=4\), and \(f(3)=3\). Thus
For the right sum, use the second endpoint of each subinterval. The heights are \(f(1)=2\), \(f(2)=4\), \(f(3)=3\), and \(f(4)=1\). Thus
The left-endpoint estimate is 9 square units, and the right-endpoint estimate is 10 square units. As a check, the left calculation uses exactly the four left-endpoint heights and the right calculation uses exactly the four right-endpoint heights; each height is multiplied by its subinterval width of 1.
Worked Example: A Midpoint Sum
Worked Example: A Midpoint Sum
A graph on \([0,6]\) is divided into three equal subintervals. Read its heights at the subinterval midpoints \(x=1\), \(x=3\), and \(x=5\) as approximately \(2\), \(4\), and \(1\), respectively. Find the midpoint rectangle sum.
There are three subintervals, each of width
The midpoint heights are \(f(1)\approx2\), \(f(3)\approx4\), and \(f(5)\approx1\). Multiply each height by the width 2 and add:
The midpoint estimate is approximately 14 square units. The check is that three rectangles have been included, their total width is \(2+2+2=6\), and each height used is read at the midpoint of its own subinterval.
Signed Rectangles and Units
When the graph crosses the horizontal axis, some rectangles may lie below it. Their heights are negative, so their signed contributions reduce the total. This is the correct interpretation when the goal is accumulated change, such as net change from a rate. If the goal is instead total geometric area, below-axis contributions must be treated as positive areas; that is a different quantity.
Units also follow from the multiplication. If the horizontal variable is measured in seconds and \(f(x)\) is measured in metres per second, then a rectangle’s width is in seconds and its height is in metres per second. The product is in metres, the unit of accumulated displacement.
Worked Example: Rectangles Above and Below the Axis
A graph is partitioned into three subintervals with widths \(2\), \(1\), and \(3\). At the selected sample locations, the graph heights are \(3\), \(-2\), and \(1\), respectively. Find the signed rectangle sum.
Multiply each signed height by its own width. The first rectangle contributes \(3(2)=6\). The second lies below the horizontal axis and contributes \((-2)(1)=-2\). The third contributes \(1(3)=3\). Therefore
The signed estimate is 7 square units. As a check, the total width is \(2+1+3=6\), and the negative rectangle has reduced the sum by 2. If the question asked for the sum of geometric areas instead, the calculation would be \(6+2+3=11\) square units; that is not the signed accumulation.
A Reliable Graph-to-Sum Procedure
Keeping the graph-reading and arithmetic steps separate makes it easier to catch mistakes. First establish the subintervals and their widths. Then read the relevant heights from the graph. Only after those values are recorded should you multiply and add.
Identify the left and right endpoints, the number of subintervals, and every subinterval width.
Use left endpoints, right endpoints, or midpoints as required. For unequal subintervals, calculate each midpoint separately.
Use the vertical scale. Keep heights below the horizontal axis negative and label approximate readings as approximate.
For each rectangle, multiply its height by its own width, then add all signed contributions.
State the rectangle rule used, whether the total is approximate, and the appropriate units.
Common Pitfall: Using the Wrong Width or Sign
A common error is to use one width for every rectangle even when the partition is unequal. In the signed-rectangle example above, multiplying all three heights by 2 would give \(3(2)+(-2)(2)+1(2)=4\). This is wrong because the widths are \(2\), \(1\), and \(3\), not \(2\), \(2\), and \(2\). The correct calculation is \(3(2)+(-2)(1)+1(3)=7\).
Another error is to turn a below-axis height into a positive number when calculating accumulated change. The graph’s position relative to the horizontal axis carries information: a negative rate produces negative change. Check each rectangle against the graph before adding, and use the width belonging to that particular subinterval.
Check Your Understanding
Use the graph information described in each question. Keep signed heights when the question asks for accumulated change.
- A graph has heights \(1,3,2\) at the three left endpoints of subintervals of width 2. What is the left-endpoint sum?
- A graph on \([0,4]\) is split into two equal subintervals. Its midpoint heights are \(2\) and \(-1\). Find the midpoint sum.
- For an interval with widths \(1\) and \(3\), the selected heights are \(-2\) and \(4\). What is the signed rectangle sum?
- Why is a height below the horizontal axis negative in a signed accumulation estimate?
- If the horizontal units are hours and the graph’s vertical units are kilometres per hour, what units does a rectangle contribution have?
