Integral of Affine Functions
Given a function and two real values, a and b (these two numbers are called limits of integration), we write the definite integral as The intuitive meaning is the area bounded between the xaxis, the graph of the function and the vertical lines x = a and x = b. In some cases that area is considered positive and in other cases is negative. We are going to see some examples later.
The integral concept is associate to the concept of area. We began considering the area limited by the graph of a function and the xaxis between two vertical lines.
The objective of this page is to play with these ideas in the simplest case of linear functions. A linear function is a polynomial of degree equal or less than 1 and its graph is a straight line. We can start with constant functions (horizontal lines). We can consider constant functions as polynomial functions of degree 0.
In this case the area is above the xaxis, then the area (the integral) is positive. In this case the area is below the xaxis, then the area (the integral) is negative. The integral of a constant function (a polynomial of degree 0) is a straight line, it is to say, a polynomial of degree 1. (The constant function f(x) = 0 is an exception). If we change the constant function you can see that the integral changes. But the only thing that change is the slope of the line. If we change the "lower" limit of integration (blue pale dot) we can see how the straight line is going up and down. When we consider a general linear function it is still very easy to calculate the integral function. In these cases we can see that some areas are positive and some areas are negative: We say that quadratic functions are integrable functions because we can calculate an indefinite integral of a parabola. The integral function of a linear function (a polynomial of degree 1) is a parabola (a polynomial of degree 2).
If we change the lower limit of integration, the integral function (the parabola) goes up and down. (Why?) Changing we lower limit of integration we get different integral functions but they are all the same up to a constant (vertical translation). The integral function of a constant function (a polynomial of degree 0) is a polynomial of degree 1. The integral function of a polynomial of degree 1 is a polynomial of degree 2. When we integrate these functions the result is a polynomial of degree one more than the original function. Remember that when we derive a linear function the result is one less than the original function. These results are related to the Fundamental Theorem of Calculus. Now we are going to study the average value of a function in this simple case of linear functions. The average value of a function f(x) over the interval [a,b] is given by
The idea is that the area under the function (positive or negative) ... ... is the same as the area of a rectangle whose height is the average value. As Linear Function are continuous, they are special cases of the Mean Value Theorem for integrals: If f(x) is a continuous function over the interval [a,b] then there is a number c in [a,b] such that REFERENCES
Michael Spivak, Calculus, Third Edition, PublishorPerish, Inc.
Tom M. Apostol, Calculus, Second Edition, John Willey and Sons, Inc.
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We can see some basic concepts about integration applied to a general polynomial function. Integral functions of polynomial functions are polynomial functions with one degree more than the original function.
As an introduction to Piecewise Linear Functions we study linear functions restricted to an open interval: their graphs are like segments.
A piecewise function is a function that is defined by several subfunctions. If each piece is a constant function then the piecewise function is called Piecewise constant function or Step function.
A continuous piecewise linear function is defined by several segments or rays connected, without jumps between them.
The Fundamental Theorem of Calculus tell us that every continuous function has an antiderivative and shows how to construct one using the integral.
The Second Fundamental Theorem of Calculus is a powerful tool for evaluating definite integral (if we know an antiderivative of the function).
The integral concept is associate to the concept of area. We began considering the area limited by the graph of a function and the xaxis between two vertical lines.
Monotonic functions in a closed interval are integrable. In these cases we can bound the error we make when approximating the integral using rectangles.
If we consider the lower limit of integration a as fixed and if we can calculate the integral for different values of the upper limit of integration b then we can define a new function: an indefinite integral of f.
The integral of power functions was know by Cavalieri from n=1 to n=9. Fermat was able to solve this problem using geometric progressions.
Lagrange polynomials are polynomials that pases through n given points. We use Lagrange polynomials to explore a general polynomial function and its derivative.
If the derivative of F(x) is f(x), then we say that an indefinite integral of f(x) with respect to x is F(x). We also say that F is an antiderivative or a primitive function of f.
Two points determine a stright line. As a function we call it a linear function. We can see the slope of a line and how we can get the equation of a line through two points. We study also the xintercept and the yintercept of a linear equation.
Power with natural exponents are simple and important functions. Their inverse functions are power with rational exponents (a radical or a nth root)
Polynomials of degree 2 are quadratic functions. Their graphs are parabolas. To find the xintercepts we have to solve a quadratic equation. The vertex of a parabola is a maximum of minimum of the function.
Polynomials of degree 3 are cubic functions. A real cubic function always crosses the xaxis at least once.
We can consider the polynomial function that passes through a series of points of the plane. This is an interpolation problem that is solved here using the Lagrange interpolating polynomial.
Archimedes show us in 'The Method' how to use the lever law to discover the area of a parabolic segment.
In his book 'On Conoids and Spheroids', Archimedes calculated the area of an ellipse. We can see an intuitive approach to Archimedes' ideas.
In his book 'On Conoids and Spheroids', Archimedes calculated the area of an ellipse. It si a good example of a rigorous proof using a double reductio ad absurdum.
Kepler was one mathematician who contributed to the origin of integral calculus. He used infinitesimal techniques for calculating areas and volumes.
