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# Week 12 Problem Set B.
## Reading.
Chapter 5.1, 5.2, 5.3.
## Problems.
### Interpreting Riemann sums as definite integrals.
In each of the following Riemann sums, try to express it as the definite integral of something, in the form $\int_{a}^{b}f(x)dx$ for some $f(x)$ and interval $[a,b]$. No need to evaluate it for now.
1. $\displaystyle \lim_{\Vert P\Vert \to 0} \sum_{k=1}^{n}c_{k}^{2} \Delta x_{k}$, where $P$ is a partition of $[0,2]$ and $c_{k}$ sample points.
2. $\displaystyle\lim_{\Vert P \Vert \to 0} \sum_{k=1}^{n}(c_{k}^{2}-3c_{k})\Delta x_{k}$, where $P$ is a partition of $[-7,5]$.
3. $\displaystyle\lim_{\Vert P \Vert \to 0} \sum_{k=1}^{n} \frac{1}{1-c_{k}}\Delta x_{k}$, where $P$ is a partition of $[2,3]$.
Now, sometimes a limit of a Riemann sum looking expression is actually the definite integral of something, especially they can be seen as a right hand end point Riemann sum (or sometimes left hand end point, or midpoint).
Try expressing the following as the definite integral of something, in the form $\int_{a}^{b}f(x)dx$ for some $f(x)$ and interval $[a,b]$. You have to think about what should be each piece. (There may be different ways of doing each)
1. $\displaystyle\lim_{n\to \infty} \sum_{k=1}^{n} \left( 3+\frac{k}{n} \right) \frac{1}{n}$
2. $\displaystyle\lim_{n\to \infty} \sum_{k=1}^{n} \left( 3+\frac{k^{2}}{n^{2}} \right) \frac{1}{n}$
3. $\displaystyle\lim_{n\to \infty} \sum_{k=1}^{n} \left( 3+\frac{k}{n} \right)^{2} \frac{1}{n}$
### Using the definite integral rules.
Suppose $f$ and $g$ are integrable and that $$
\int_{1}^{2}f(x)dx = -4 \,\,,\int_{1}^{5}f(x)dx=6\,\,,\int_{1}^{5}g(x)dx=8.
$$
Using the rules for integrals to find the following:
1. $\displaystyle\int_{2}^{2}g(x)dx$
2. $\displaystyle\int_{5}^{1} g(x)dx$
3. $\displaystyle \int_{1}^{2} 3f(x)dx$
4. $\displaystyle\int_{2}^{5}f(x)dx$
5. $\displaystyle\int_{1}^{5}[f(x)-g(x)]dx$
6. $\displaystyle\int_{1}^{5}[4f(x)-g(x)]dx$
Suppose $h$ is integrable, and that $\int_{0}^{3}h(z)dz=3$ and $\int_{0}^{4}h(z)dz=7$, find the following:
1. $\int_{0}^{3}h(u)du$
2. $\int_{3}^{4}h(t)dt$
3. $\int_{4}^{3}h(y)dy$
### Using known areas to evaluate definite integrals.
In each of the following, graph the integrands and use areas to evaluate the integrals.
1. $\displaystyle\int_{-2}^{4}\left( \frac{x}{2}+3 \right)dx$
2. $\displaystyle\int_{-3}^{3}\sqrt{9-x^{2}}dx$
3. $\displaystyle \int_{-2}^{1}|x|dx$
4. $\displaystyle \int_{-1}^{1}(2-|x|)dx$
5. $\displaystyle\int_{-1}^{1}(1+ \sqrt{1-x^{2}}dx$
### Some specific formulas for definite integrals.
We will eventually develop methods (using antiderivatives and the Fundamental theorem of calculus) to help us evaluate definite integrals. But let us develop some formulas directly.
1. We have definite integral $$
\int_{a}^{b}x dx = \frac{b^{2}}{2}-\frac{a^{2}}{2}
$$ Show this formula is true by (1) first setting up a right hand endpoint Riemann sum with $n$ pieces for this integral, and then (2) take the limit as $n\to\infty$. You may need one of the Faulhaber formulas. Also, we know we can just use the right hand endpoint Riemann sum because $x$ is continuous so $x$ is Riemann integrable, and any Riemann sum would converge to the same.
2. We have definite integral $$
\int_{a}^{b} x^{2}dx = \frac{b^{3}}{3}-\frac{a^{3}}{a^{3}}
$$ Show this formula is true by (1) first setting up a right hand endpoint Riemann sum with $n$ pieces for this integral, and then (2) take the limit as $n\to\infty$.
3. Now using these two results, and any of the integral rules, evaluate each of the following:
1. $\displaystyle\int_{\sqrt{2}}^{2} x dx$
2. $\displaystyle\int_{0}^{0.3} t^{2}dt$
3. $\displaystyle\int_{2}^{1}(1+\frac{z}{2})dx$
4. $\displaystyle\int_{-4}^{2} (s+\frac{s^{2}}{4})ds$
5. $\displaystyle\int_{3}^{7} (3x^{2}-2x+1)dx$
### Finding average values.
In each of the following, sketch the function $f$ over the interval $[a,b]$, and find the average value $av_{[a,b]}f$ over the given interval.
1. $f(x) = x^{2}-1$ on $[0,\sqrt{3}]$
2. $f(x) = -3x^{2}-1$ on $[0,1]$
3. $f(t)=(t-1)^{2}$ on $[0,3]$
4. $f(x)=|x|-1$ on $[1,3]$
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