With geometry, the lines involved in polygons are straight so it’s easy to determine the area. However, in some instances, the lines on a graph will be curved instead of straight. These are the representation of a function on the graph. When an area needs to be found on the graph of a function, the formulas used to find the area of a polygon will not work because they are not precise when a curved line is involved. Instead, calculus is used to find the exact area.
The Fundamental Theorem of Calculus
To find the area of a space on a graph based on a function, it’s necessary to use the fundamental theorem of calculus. In this theorem, it’s assumed there is a function on the graph and that there are two points denoting the beginning and end of the area to be measured. The space between these two points is the integral.
The theorem can be split into two different parts. The first part denotes what is needed and the second part is simply how to solve it. Written out, the theorem states the integral from a to b of f(x)dx is equal to g(b)-g(a) where g is the antiderivative of f(x). This simply means the area is found by subtracting the antiderivative of the function solved for a from the antiderivative of the function solved for b.
To solve the theorem for a function, it’s necessary to find the antiderivative for that function. This becomes g(x). Then, a and b are plugged in to this and solved. Finally, the answer for a is subtracted from b to get the area that is needed.
Finding the Antiderivative
One of the most important parts of solving the fundamental theorem of calculus is understanding how to find the antiderivative of the function. The way to find the antiderivative depends on the type of function in the problem. With functions that have an exponent, for example, the power rule will be applicable. In many cases, the reverse rules will be needed to find the antiderivative of the function.
Why Calculus is Easy
Calculus is made up of only two parts: differentials and integrals. Differential calculus basically involves finding derivatives and integral calculus in general involves finding integrals. Finding derivatives involves dividing and finding integrals involves multiplying. This is the basics of calculus and so long as a person understands the algebra and trig formulas needed to do the above, they’ll find it’s easy to do many different calculus problems.
Although it’s easy to find the area when all the lines are straight, this is not possible in all cases. Calculus and the fundamental theorem of calculus allow a student to find the area in graphs that include a function instead of a straight line. This is a basic component to calculus and, once learned, can enable the student to do many calculus problems easily. One of the most common examples of this in real world use is to use the fundamental theorem to determine how fast a car was going at a precise time, not just the average of how fast they were going during the entire trip. Since the car can speed up and slow down, the graph would include a curved line so the fundamental theorem is needed as the formulas used for polygons are not going to provide a precise answer.