6.2 Areas and Riemann Sums

This section begins to form the link between antiderivatives and the area under a curve.

Area Under a Graph If f(x) is a continuous nonnegative function on the interval we refer to the area of the region shown as the area under the graph of f(x) from a to b.

The computation of the area is not trivial when the top boundary of the region is curved. However, we can estimate the area to any desired degree of accuracy. How?

We can construct rectangles whose combined areas are approximately the same as the area to be computed.

Constructing the Rectangles Given a continuous nonnegative function f(x) on the interval 1.Divide the x-axis into n equal subintervals, where n is some positive integer. This subdivision is called a partition of the interval from a to b. The width of the entire interval is b – a, so the width of each subinterval is (b – a)/n. We will denote this width as Δx. Δx = (b – a)/n

Constructing the Rectangles 2. In each subinterval, select a point (any point in the subinterval will do). Let x1 be the point in the first interval, x2 be the point in the second, etc. These points are used to form the rectangles that approximate the area under the graph of f(x).

Constructing the Rectangles 3. Construct the first rectangle with height f(x1) and the first subinterval as the base. The top of the rectangle touches the graph directly above x1. Then, [area of first rectangle] = [height][width] = f(x1) Δx The second rectangle rests on the second interval and has height f(x2). Its area is [area of second rectangle] = [height][width] = f(x2) Δx

Constructing the Rectangles 4. If we continue in this fashion, our estimate of the area under the graph will be given by summing the area of these n rectangles [area estimate] = f(x1) Δx + f(x2) Δx +…+ f(xn) Δx [area estimate] = [f(x1) + f(x2) +…+ f(xn)] Δx This sum is called a Riemann sum. It provides an approximation to the area under the graph of f(x) when f(x) is nonnegative and continuous.

As the number of subintervals increases indefinitely, the Riemann sums approach a limiting value…the area under the graph.

Notes Regarding the Area Under a Graph There is a connection between the rate of change of a function and the amount of increase of the function. If a quantity is increasing, then the area under the rate of change function from a to b is the amount of increase in the quantity from a to b.

Interpretation of Areas Examples Functiona to bAreas under the Graph from a to b Marginal cost at production level x 30 to 60Additional cost when production is increased from 30 to 60 units Rate of sulfur emissions from a power plant t years after to 3Amount of sulfur released from 1991 to 1993 Birth rate t years after 1980 (in babies per year) 0 to 10Number of babies born from 1980 to 1990 Rate of gas consumption t years after to 6Amount of gas used from 1988 to 1991