The answer to why a thing doesn't matter is often answered by how it does matter. The lower limit of integration in the FTC is a good example of this. It also helps students to understand these relationships if they are related to mathematical concepts which act as a solid foundation for the Calculus. The solid foundation for the Calculus is not what students studied during the last semester. It's the algebra and geometry they've been learning for the past four years.
It helps to draw pictures for the graphs described below.
For simplicity consider the graph of a continuous function, y(x), in the first quadrant. Denote the integral with a more algebraic notation: the area from a to x of the function y, as A_a,x( y ) . We can write the equation of the tangent line to the area function at a point x=b using the point-slope form for the equation of a straight line.
A_t = m(x-b) + A_a,b( y )
The first term, m(x-b), has the form of the area of a rectangle. And since we know that the constant component of the boundary function produces the linear component of the area function, the slope, m, is just the height of the boundary function. m=y(b).
So in considering the tangent line to the area graph at x=b,
A_t = y(b)(x-b) + A_a,b( y )
the lower limit of integration enters through the value of area function at the point of tangency,
( b, A_a,b( y ) )
and the slope of the tangent line, m( A( y(x) ) )=y(x), is the central content of the FTC.
In a message dated 2/15/2012 8:57:01 A.M. Eastern Standard Time, firstname.lastname@example.org writes:
Can someone please explain to me, in layman's terms, why the lower limit of integration doesn't matter when using the Fundamental Theorem of Calc to evaluate the definite integral as a function? I get the "what" of the FTC, but I can't get a handle on the "why" of the irrelevance of the value of a.