There is one important type of problems in which we know the derivative of a function (its rate of change, the slope of its graph) and we want to find the function. For example, we know velocity and want to calculate the space. The process of finding a function from its derivative is called antidifferentiation, finding a primitive function, or finding an indefinite integral. As the name implies, antidifferentiation is an inverse operation to differentiation. The use of the word integration here could seem odd because the problem of integration is related in some way with the finding of an area (accumulation, summation) and differentiation is related with the instantaneous rate of change or as the slope of the tangent line to the graph of the function. We are going to see later then this two very different problems are deeply connected (Fundamental Theorem of Calculus) and that, in some way, integration is the inverse process of differentiation. F(x) is said to be an antiderivative (or a primitive or an indefinite integral) of f(x) on an open interval if the derivative of F is f for every value of x on the interval. Notice that we define an antiderivative and not the antiderivative. This is because an antiderivative is not unique. Nevertheless the antiderivative is unique up to an additive constant. It is say: Any two antiderivatives F and G of the same function f can differ only by a constant. The reason is because their difference FG has the derivative 0 and then FG is a constant function.
Sometimes we use a particular sign (from Leibniz, the sign for integration is a large 's' from 'sum'. Leibniz used the following sign to denote a general primitive of f): The function f is called the integrand, the constant C is called the constant of integration. The symbol dx indicates that we are to integrate with respect to x. If we know simple techniques of differentiation to find some antiderivatives is easy. For example, it is straightforward to find a primitive for a constant function: We can check that in each point the derivative is the original constant function: The derivative of a polynomial of degree 1 (linear function) is a constant function (degree 0, an horizontal line). Then, an antiderivative of a constant function is a linear function. Antiderivatives of polynomials functions are easy too. For example, a basic linear function (the identity function): Checking the result in one point: The derivative of a polynomial of degree 2 (a parabola) is a polynomial of degree 1 (a linear function). Then, a primitive of linear function is a parabola. A quadratic polynomial (a parabola): The derivative of a polynomial of degree 3 (a cubic) is a polynomial of degree 2 (a parabola). Then, an antiderivative of a parabola is a cubic function. Another example, a cubic polynomial:
In the next version of the interactive application you can move the blue dot along the xaxis: When we differentiate a polynomial function we get a polynomial function of one degree less than the original function. When we find an antiderivative of a polynomial function we get a polynomial function of one degree more than the original function. Although finding primitives of polynomial functions is easy, to find an antiderivative turns out to be a difficult problem in general. The Fundamental Theorem of Calculus tell us that we can always construct an antiderivative (primitive) of a continuous function by integration. REFERENCES
Michael Spivak, Calculus, Third Edition, PublishorPerish, Inc.
Tom M. Apostol, Calculus, Second Edition, John Willey and Sons, Inc.
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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.
To calculate the area under a parabola is more difficult than to calculate the area under a linear function. We show how to approximate this area using rectangles and that the integral function of a polynomial of degree 2 is a polynomial of degree 3.
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.
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).
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 integral of power functions was know by Cavalieri from n=1 to n=9. Fermat was able to solve this problem using geometric progressions.
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.
