integration

Question

2. Relate the process of integration to the area under a curve and state the function of the constant of integration. With and illustration, determine the integral of a simple algebraic expression

Answer 1

Richard,

 

Let f(x) be any continuous function of x, any real number.  And f(x) = x for our simple example.

     F(x) be its indefinite integral function with respect to x.

        C  be a constant of integration for an indefinite integral.

     a, b (b≥a) be the beginning and ending values of x for a definite integral of x over interval [a,b]. 

 

Imagine building a wall out of slats, whose width was always the same Δx, but whose height would be cut to the value of the function f(x), and each slat would be nailed in vertically centered on its horizontal position x (with no space allowed between the slats).  Then the changing height of this wall of slats would resemble a graph of f(x) vs x, with some small portions of each slat top slightly above and below the smoothed curve.  If you used narrower slats, those small error portions would get even smaller, and disappear altogether as the width of the slat approached zero in the limit. 

 

Also, each slat is a narrow rectangle that has an area given by its height times width : A(x) = [f(x) Δx].  This is the area contribution of the slat centered at x.  Then the area of the entire slat wall is just the sum of all these slat areas:

 

Area = ∑[f(x) Δx] over all slat pieces, but this area still includes all those little errors at the top pf each slat.  To eliminate them, we shrink Δx to the limit and replace the sigma discrete addition sign with the integral sign.  Then integration becomes finding the true area under the function f(x).

 

True area = LIMIT over all Δx as Δx→0 of ∑[f(x) Δx]

              = ∫f(x)dx   =  F(x) + C  , where in the limit  Δx becomes dx (d for delta).

 

In an indefinite integral, when no a & b values are specified:

The x in F(x) is the implied endpoint b , and C substitutes for the unspecified starting point.  Once a definite starting point is specified, the constant C cancels out.  Thus, with a definite integral over [a,b) or [a,x]:

 

∫_a^b f(x)dx = [F(x) + C]|^x=b  -   [F(x) + C]|_x=a  =  [F(b) + C] - [F(a) + C] = F(b) - F(a).  Note, b = x if not specified.

 

From this, you can also see that integrating from a to a is zero, integrating from b to a is the negative of integrating from a to b.  And also that the derivative of [F(x) + C] = [F'(x) + C'] = F'(x) = f(x) since C'=0 being a constant.  So it all hangs together.

 

In our example, f(x) = x, ∫xdx = 0.5x² + C  for the indefinite integration with no a & b.

If a =0 , then ∫xdx = 0.5x² + C and F(x) = 0.5x² since the C's cancel

If a = 2, then ∫xdx = [0.5x² + C]|^x  -  [0.5(2)² + C]|_x=2 = [0.5x² - 2] = F(x)

and if further b=3 (with a=2), then F(3) = 4.5 - 2 = 2.5