Area Under the Curve
Start with the area under a simple curve, a straight line. A straight line, y = const, over a distance in x is a rectangle and the area of a rectangle is the height times the width. For a little more excitement look at the area under, or within, a triangle formed by the coordinate axes and y = kx. These two areas, the rectangle and the triangle, have straight sides, but what about an area bounded by a quadratic, y = x2. All of a sudden this is a whole new problem. As a first approach to this problem look at successive approximations to the area.
| Graph the function between x = 0 and x = 2. The area under this curve is less than the area within a triangle formed with base along the x axis from 0 to 2, height from y = 0 to 4 and the slant height from the point (0,0) to (2,4). Such a triangle has area (1/2)2·×4 = 4. This is a first approximation. |
 |
| The
curve y = x2 goes through the points (0.0),
(1,1) and (2,4) so continue this approximation approach by finding the
area of this triangle and trapezoid combination. The area of the triangle is (1/2)1×2
= 1/2. The area of a trapezoid is (1/2)(sum of the opposite
faces)·(height) which for this trapezoid is
(1/2)(1 + 4)(1) = 2.5. The sum of these
areas is 3. This is a better approximation. |
 |
This approximation approach can be carried to greater accuracy by making smaller and smaller trapezoids. Make the trapezoids small enough and they get pretty close to rectangles. Also note that this approximation approach can be applied to any function; power, exponential, trigonometric or any combination thereof. Thus we have an approximation method that can be carried out to any degree of accuracy as long as we are willing to make the detailed calculations.
With this introduction to areas and an approximation approach to areas bounded by functions look now at what is called the fundamental theorem of integral calculus.
Use the same curve y = x2 as an example, though any curve would work as well, and look to approximating the area not with trapezoids, but with a collection of narrow rectangles. The rectangles can be constructed in a variety of ways, inside the curve, outside the curve or using a mid-value. It really doesn't make any difference how they are constructed because we are going to take the limit by making their width go to zero. The ones shown here are an average height. Look at the xn’th rectangle of width Dx that has height xn2.
The area under this curve can be written as a sum of similar rectangles. With this view, the area under the curve is
with the area getting closer and closer to the actual area as the width of the rectangles decreases and their number increases.
Using a limit approach, and the knowledge that this summation, or integral, over a specified range in x is the area under the curve, A is the limit of the sum as Dx goes to zero.
The integral is viewed as the area generated by summing an infinite number of rectangles of infinitely small width. The antiderivatives of these integrals can be shown by the approximation method outlined to be equal to the area under the curve. In actual practice the integrals are usually specified not as 0 to x but as from a lower limit to a higher limit equivalent to the area desired.
The next three problems illustrate the use of integrals in finding the area of a rectangle, triangle and area under a quadratic. These integrals demonstrate that the area under the curve as represented by the integral is actually the antiderivative with the given limits. This is easily proven by performing approximations as outlined above.
The problems listed below are typical area calculation problems. Some are simple area calculations while some involve "negative" areas, and some are business or health oriented.
TAUC05 Simple area calculation
AUC06 Area calculation with symmetry
AUC20 Positive and negative areas
AUC40 Total return on investment
AUC41 Growth of a piece of porcelain