MB
Matt B.

### Problem 41

The following table gives the percent growth of the U.S. population beginning in July of the year indicated. If the U.S. population was 281,421,906 in July 2000, estimate the U.S. population in July 2010. Darren M.

### Problem 42

In the following exercises, estimate the areas under the curves by computing the left Riemann sums, $L_{8} .$

MB
Matt B.

### Problem 43

In the following exercises, estimate the areas under the curves by computing the left Riemann sums, $L_{8} .$ Darren M.

### Problem 44

In the following exercises, estimate the areas under the curves by computing the left Riemann sums, $L_{8} .$

MB
Matt B.

### Problem 45

In the following exercises, estimate the areas under the curves by computing the left Riemann sums, $L_{8} .$ Darren M.

### Problem 46

[IT] Use a computer algebra system to compute the Riemann sum, $L_{N}, \quad$ for $\quad N=10,30,50$ for $f(x)=\sqrt{1-x^{2}}$ on $[-1,1]$

MB
Matt B.

### Problem 47

[T] Use a computer algebra system to compute the Riemann sum, $L_{N}, \quad$ for $\quad N=10,30,50$ for $f(x)=\frac{1}{\sqrt{1+x^{2}}}$ on $[-1,1]$ Darren M.

### Problem 48

[T] Use a computer algebra system to compute the Riemann sum, $L_{N},$ for $N=10,30,50$ for $f(x)=\sin ^{2} x$ on $[0,2 \pi] .$ Compare these estimates with $\pi .$

MB
Matt B.

### Problem 49

In the following exercises, use a calculator or a computer program to evaluate the endpoint sums $R_{N}$ and $L_{N}$ for $N=1,10,100$ . How do these estimates compare with the exact answers, which you can find via geometry?
[T]$y=\cos (\pi x)$ on the interval $[0,1]$ Darren M.

### Problem 50

In the following exercises, use a calculator or a computer program to evaluate the endpoint sums $R_{N}$ and $L_{N}$ for $N=1,10,100$ . How do these estimates compare with the exact answers, which you can find via geometry?
$[T] \mathrm{y}=3 x+2$ on the interval $[3,5]$

MB
Matt B.

### Problem 51

In the following exercises, use a calculator or a computer program to evaluate the endpoint sums $R_{N}$ and $L_{N}$ for $N=1,10,100$ .
[T] $y=x^{4}-5 x^{2}+4$ on the interval $[-2,2]$ which has an exact area of $\frac{32}{15}$ Darren M.

### Problem 52

In the following exercises, use a calculator or a computer program to evaluate the endpoint sums $R_{N}$ and $L_{N}$ for $N=1,10,100$ .
[T] $y=\ln x$ on the interval $[1,2],$ which has an exact area of $2 \ln (2)-1$

MB
Matt B.

### Problem 53

Explain why, if $f(a) \geq 0$ and $f$ is increasing on $[a, b],$ that the left endpoint estimate is a lower bound for the area below the graph of $f$ on $[a, b]$ Darren M.

### Problem 54

Explain why, if $f(b) \geq 0$ and $f$ is decreasing on $[a, b],$ that the left endpoint estimate is an upper bound for the area below the graph of $f$ on $[a, b]$

MB
Matt B.

### Problem 55

Show that, in general, $R_{N}-L_{N}=(b-a) \times \frac{f(b)-f(a)}{N}$ Darren M.

### Problem 56

Explain why, if $f$ is increasing on $[a, b],$ the error between either $L_{N}$ or $R_{N}$ and the area $A$ below the graph of $f$ is at most $(b-a) \frac{f(b)-f(a)}{N}$

MB
Matt B.

### Problem 57

For each of the three graphs:
a. Obtain a lower bound L(A) for the area enclosed by the curve by adding the areas of the squares
enclosed completely by the curve.
b. Obtain an upper bound U(A) for the area by adding to L(A) the areas B(A) of the squares
enclosed partially by the curve. Darren M.

### Problem 58

In the previous exercise, explain why L(A) gets no smaller while U(A) gets no larger as the squares are
subdivided into four boxes of equal area.

MB
Matt B.
A unit circle is made up of n wedges equivalent to the inner wedge in the figure. The base of the inner triangle is 1 unit and its height is $\sin \left(\frac{\pi}{n}\right) .$ The base of the outer triangle is $B=\cos \left(\frac{\pi}{n}\right)+\sin \left(\frac{\pi}{n}\right) \tan \left(\frac{\pi}{n}\right)$ and the height is $H=B \sin \left(\frac{2 \pi}{n}\right)$. Use this information to argue that the area of a unit circle is equal to $\pi .$ 