Last updates: 07 December 2009
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. |
[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)