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Suppose $f$ integrable\hfil \break
 (Note $f$ continuous implies $f$ integrable).
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Suppose $a = x_0 < x_1 < x_2 < .... x_{n-1} < x_n = b$, $\Delta x = x_i - x_{i-1}$ 
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 \centerline{$\int_a^b f(x) dx = lim _{n \rightarrow \infty} \Sigma_{i = 1}^n f(x_i^*) \Delta x$ }
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If  $n$ equal subdivisions:  $\Delta x = {b - a \over n}$ and if we use right-hand endpoints: $x_i^* = x_i$
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 \centerline{$\int_a^b f(x) dx = lim _{n \rightarrow \infty} \Sigma_{i = 1}^n f(x_i) {b - a \over n}$ }
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The Fundamental Theorem of Calculus:  Suppose $f$ continuous on $[a, b]$.

1.)  If $g(x) = \int_a^x f(t) dt$, then $g'(x) = f(x)$.

2.)   $\int_a^b f(t) dt = F(b) - F(a)$ where $F$ is any antiderivative of $f$, that is $F' = f$.

The Fundamental Theorem of Calculus:  Suppose $f$ continuous on $[a, b]$.

1.)  If ${d \over dx}[ \int_a^x f(t) dt] = f(x)$.

2.)   $\int_a^b F'(t) dt = F(b) - F(a)$.

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Examples:
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1.)  If $g(x) = \int_0^x t^2 dt$, then $g'(x) = \underline{\hskip 1in}$.

2.)  If $g(x) = \int_5^x t^2 dt$, then $g'(x) = \underline{\hskip 1in}$.

3.)  If $g(x) = \int_-2^x sin(t^2) dt$, then $g'(x) = \underline{\hskip 1in}$.


4.)  If $g(x) = \int_4^x tan({t^3 \over t+ 1}) dt$, then $g'(x) = \underline{\hskip 1in}$.


5.)  If $g(x) = \int_1^x \sqrt{3t - 5} dt$, then $g'(x) = \underline{\hskip 1in}$.

Examples:
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1.)  If $g(x) = \int_0^{x^3} t^2 dt$, then $g'(x) = \underline{\hskip 1in}$.

2.)  If $g(x) = \int_5^{x^3} t^2 dt$, then $g'(x) = \underline{\hskip 1in}$.

3.)  If $g(x) = \int_2^{ln(x)} {t \over t+1} dt$, then $g'(x) = \underline{\hskip 1in}$.

4.)   If $g(x) = \int_3^{1 \over x} sec(t) dt$, then $g'(x) = \underline{\hskip 1in}$.

Evaluate the limit by recognizing the sum as a Riemann sum for a function defined on [0, 1]

1.)  $lim _{n \rightarrow \infty} \Sigma_{i = 1}^n sin({i \over n}){1 \over n}$ 

2.)  $lim _{n \rightarrow \infty} \Sigma_{i = 1}^n {i^5 \over n^6}$ 

%%3.)  $lim _{n \rightarrow \infty} \Sigma_{i = 1}^n sin({i \over n}){1 \over n}$ 



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