We introduce the basic idea of using rectangles to approximate the area under a curve.
Rectangles and areas
We want to compute the area between the curve and the horizontal axis on the interval :
As we add rectangles, we are more closely approximating the area we are interested in:
Let’s setup some notation to help with these calculations:
But which set of rectangles?
When we use rectangles to compute the area under a curve, the width of each rectangle is . It is clear that , for .
But how do we determine the height of the rectangle?
We choose a sample point and evaluate the function at that point. The value determines the height of a rectangle.
Rectangles defined by left-endpoints
We can set the rectangles up so that the sample point is the left-endpoint.
Rectangles defined by right-endpoints
We can set the rectangles up so that the right-endpoint determines the height.
Rectangles defined by midpoints
We can set the rectangles up so that the midpoint of the base determines the height.
Riemann sums and approximating area
Once we know how to identify our rectangles, we can compute approximations of some areas. If we are approximating area with rectangles, then
A Riemann sum computes an approximation of the area between a curve and the -axis on the interval . It can be defined in several different ways. In our class, it will be defined via left-endpoints, right-endpoints, or midpoints. Here we see the explicit connection between a Riemann sum defined by left-endpoints and the area between a curve and the -axis on the interval :
Left Riemann sums
and we find
Right Riemann sums
and we find
Midpoint Riemann sums
and we find
Consider the function on the interval . We will approximate the area between the graph of and the -axis on the interval . See the figure below.
The image depicts a LeftRightMidpoint Riemann sum with subintervals.
This approximation is an overestimateunderestimate .
Consider the function on the interval . We will approximate the area between the graph of and the -axis on the interval using a right Riemann sum with rectangles. First, determine the width of each rectangle. Next, we will determine the grid-points.
For a right Riemann sum, for , we determine the sample points as follows: Now, we can approximate the area with a right Riemann sum.
We can now simplify the last sum by using the distribution, commutativity and associativity properties of a sum.
In the last sum, the constant is a common factor, so we can again apply the distribution property and obtain the following In the first sum above, the constant is added times, and we have a formula for the second sum. Recall: . Therefore, We can simplify this expression and obtain This approximation is an overestimateunderestimate .
Riemann sums approximate the area between curves and the -axis via rectangles. When computing this area via rectangles, there are several things to know:
- What interval are we on? In our discussion above we call this .
- How many rectangles will be used? In our discussion above we called this .
- What is the width of each individual rectangle? In our discussion above we called this .
- What points will determine the height of the rectangle? In our discussion above we called these sample points, , and they can be left-endpoints, right-endpoints, or midpoints.
- What is the actual height of the rectangle? This will always be .
- We approximate the area with a Riemann sum
- As gets bigger and bigger, gets smaller and smaller, and approximation
gets better and better. We compute the exact value of by taking the limit
of Riemann sums