Problem Set - The Fundamental Theorem of Calculus

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 07 December 2009

The Fundamental Theorem of Calculus

	What does $${\int }_a^{b}f\left(x\right)dx$$ mean?
	How does one usually calculate $${\int }_a^{b}f\left(x\right)dx\text{ ?}$$ Give an example which shows that this method does not always work. Why doesn't it?
	Give an example which shows that $${\int }_a^{b}f\left(x\right)dx$$ is not always the true area under $f\left(x\right)$ between $a$ and $b$ even if $f\left(x\right)$ is contunuous between $a$ and $b$.
	What is the Fundamental Theorem of Calculus?
	Let $f\left(x\right)$ be a function which is continuous and let $A\left(x\right)$ be the area under $f\left(x\right)$ from $a$ to $x$. Compute the derivative of $A\left(x\right)$ by using limits.
	Why is the Fundamental Theorem of Calculus true? Explain carefully and thoroughly.
	Give an example which illustrates the Fundamental Theorem of Calculus. In order to do this, compute an area by summing up the areas of tiny boxes and then show that applying the Fundamental Theorem of Calculus gives the same result.

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)