Integrals

In this chapter, we deal with the question of how to determine a function if we know its derivative. So, for example, given a marginal cost function, we want to know its corresponding total cost function. Or, given a marginal revenue function, we want to know its corresponding total revenue function. The set of all functions F ( x ) $$F(x)$$ that have the property that their derivative is equal to some function f ( x ) $$f(x)$$ is called an indefinite integral. For a given function f ( x ) $$f(x)$$ , its indefinite integral is obtained by a procedure that is opposite to the procedure of finding its derivative, and therefore F ( x ) $$F(x)$$ is also called an antiderivative or a primitive function. We show three methods of integration: direct integration, integration by substitution, and integration by parts. In addition to the term indefinite integral, there is also the term definite integral, where the definite integral of a function f ( x ) $$f(x)$$ is equal to the signed area enclosed by the graph of the function f ( x ) $$f(x)$$ and the x-axes on an interval [ a , b ] $$[a,b]$$ . Thus, for example, for a given marginal cost function, the total cost at the production level q is equal to the area under the marginal cost function on the interval [ 0 , q ] $$[0,q]$$ . We apply the concept of the definite integral to the problem of finding the area enclosed by some curves and conclude by improper integrals, that is, integrals in which either the upper or lower bound of integration, or both, are not finite numbers.