Esquema para calcular el área entre curvas

Scheme for Calculating the Area between Curves

Problem Statement: A region in the plane will be described by stating curves that form the boundary of the region. There may be several curves, including straight lines. The problem is to find the area of the described region.

Step 1: Sketch the region in the plane under consideration.

If the curves are not familiar, use the scheme for sketching the graph of a function.

Step 2: From the sketch in Step 1, identify an interval on either the x-axis or y-axis that “supports” the region.

This interval will be the interval of integration and the axis on which the interval lies will correspond to the variable of integration, i.e., dx (or dy).

Step 3: Divide the interval in Step 2 into n subintervals of equal length. Using the right-hand endpoints as sample points, construct a rectangle on a typical subinterval that approximates the area of the region for that subinterval.

The area of this rectangle will approximate the area between the curves on the subinterval. The approximation of the total area will be the sum of these areas. Label the graph and points properly.

Step 4: Using the results of Step 3, calculate the area of this rectangle (and simplify when appropriate) and write the Riemann sum that approximates the total area for the region.

The area ( ) of the rectangle is given by , where the base ( ) is always the length of the subinterval ( or ) and the height ( ) is always the length of the other side of the rectangle. In calculating the area of a typical rectangle, the same formula for the area must hold for every rectangle. If this is not the case for the interval you have chosen, return to Step 2 and chose the interval to lie on the other axis. Otherwise you will have to divide the problem into more than one problem depending on the number of different ways of computing the area.

Step 5: Take the limit as of the Riemann sum found in Step 4 and interpret the limit as a definite integral.

Remember the interval of integration and variable of integration were determined in Step 2.

Step 6: Evaluate the integral.

Steps 1 – 5 will be referred to as “setting up an integral”. If you are asked to simply set up an integral, you need not perform Step 6.

Example 1: Find the area bounded by the curves and

Solution: (Figure 1 below contains the results of Steps 1 -3. The steps 1-3 described below illustrate how Figure 1 is obtained. In all the homework and exam problems I expect to see such a figure.)

Step 1: (It is essential that the geometry is correct. The second function is a straight line with slope one-half and y-intercept . The first function is a translation of the basic square root function . So the only real question is where the curves intersect.)

To find the points of intersection, set .

For : . So one point of intersection is .

For : . So the other point of intersection is .

Step 2: (You are looking for an interval that supports the area show in Step 1. Project the area onto the x-axis and observe the interval. Your answer is the interval of integration.)

The interval that supports the area is on the x-axis.

Step 3: (Choose a few points, but more than two or three, and construct the rectangles that approximate the area on one of the resulting subintervals. In Figure 1, there are seven subintervals. Notice that the relevant points are labeled.)

Step 4: (From the geometry of Figure 1, write the area of the rectangle drawn and simplify the resulting expression. Then add all these areas together.)

(an approximation of the area between the curves.)

Step 5:

Step 6:

Example 1 can also be solved using an interval on the y-axis rather than an interval on the x-axis. The following example shows how to do this.

Example 2: Find the area bounded by the curves and.

Solution: (Figure 2 below contains the results of Steps 1 -3. The steps 1-3 described below illustrate how Figure 2 is obtained. In all the homework and exam problems I expect to see such a figure.)

Step 1: (Again one must get the geometry correct. The same analysis as in Example 1 can be used.)

Step 2: (You are looking for an interval that supports the area show in Step 1. This time project the area onto the y-axis instead of the x-axis as was done in Example 1. Your answer is the interval of integration.)

The interval that supports the area is on the y-axis.

Step 3: (Choose a few points, but more than two or three, and construct the rectangles that approximate the area on one of the resulting subintervals. In Figure 2, there are eight subintervals. Notice that the relevant points are labeled. In labeling the graphs, it is helpful to write the equations of the curves as functions of y rather than functions of x as they are stated in the problem.)

Step 4: (From the geometry of Figure 2, write the area of the rectangle drawn and simplify the resulting expression. Then add all these areas together.)

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