Section 1
Preparation For The Definite Integral
In Exercises $1-3$ compute $L_{f}(P)$ and $U_{f}(P)$ for the indicated partition.
Compute $L_{f}(P)$ and $U_{f}(P)$ for the indicated partition.
In Exercises $4-6$ compute the left sum and the right sum for the indicated partition.
Compute the left sum and the right sum for the indicated partition.
In Exercises $7-12$ compute $L_{f}(P)$ and $U_{f}(P)$.$f(x)=x+2 ; P=\left\{-1,-\frac{1}{2}, 0, \frac{1}{2}, 1, \frac{3}{2}, 2\right\}$
Compute $L_{f}(P)$ and $U_{f}(P)$.$f(x)=x^{2} ; P=\left\{-1,-\frac{1}{2}, 0, \frac{1}{2}, 1, \frac{3}{2}, 2\right\}$
Compute $L_{f}(P)$ and $U_{f}(P)$.$f(x)=-1 / x ; P=\{-4,-3,-2,-1\}$
Compute $L_{f}(P)$ and $U_{f}(P)$.$f(x)=\sin x ; P=\{0, \pi / 6, \pi / 4, \pi / 3, \pi / 2\}$
Compute $L_{f}(P)$ and $U_{f}(P)$.$f(x)=\sin x+\cos x ; P=\{0, \pi / 4, \pi / 2\}$
Compute $L_{f}(P)$ and $U_{f}(P)$.$f(x)=x^{4}-2 x^{2} ; P=\{-2,-1,0,1,2\}$
In Exercises $13-16$ use a calculator to approximate $L_{f}(P)$ and $U_{f}(P)$$f(x)=\sqrt{x} ; P=\left\{0, \frac{1}{9}, \frac{2}{9}, \ldots, \frac{8}{9}, 1\right\}$
Use a calculator to approximate $L_{f}(P)$ and $U_{f}(P)$$f(x)=\sqrt{1-x^{2}} ; P=\left\{-1,-\frac{3}{4},-\frac{1}{2},-\frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1\right\}$
Use a calculator to approximate $L_{f}(P)$ and $U_{f}(P)$$f(x)=\ln x ; P=\{0.5,0.75,1,1.25,1.5,1.75,2\}$
Use a calculator to approximate $L_{f}(P)$ and $U_{f}(P)$$f(x)=e^{-2 x}-1 ; P=\{-1,-0.8,-0.6,-0.4,-0.2,0,0.2,0.4,0.6,0.8,1\}$
In Exercises $17-20$ compute the left sum, right sum, and midpoint sum for the given function and partition.$f(x)=2 x-3 ; P=\{-2,-1,0,1\}$
Compute the left sum, right sum, and midpoint sum for the given function and partition.$f(x)=2 x^{2}-1 ; P=\left\{-1,0, \frac{1}{2}, 1\right\}$
Compute the left sum, right sum, and midpoint sum for the given function and partition.$f(x)=1 / x^{2} ; P=\left\{1, \frac{3}{2}, 2, \frac{5}{2}, 3\right\}$
Compute the left sum, right sum, and midpoint sum for the given function and partition.$f(x)=\sin x ; P=\{0, \pi / 2, \pi, 2 \pi\}$
In Exercises 21-24 use a calculator to compute the left sum, right sum, and midpoint sum for the given function and interval, using a partition having the indicated number of subintervals of the same length.$f(x)=\sqrt{4+x^{2}} ;[-1,1] ; n=10$
Use a calculator to compute the left sum, right sum, and midpoint sum for the given function and interval, using a partition having the indicated number of subintervals of the same length.$f(x)=\sin x ;[0, \pi] ; n=50$
Use a calculator to compute the left sum, right sum, and midpoint sum for the given function and interval, using a partition having the indicated number of subintervals of the same length.$f(x)=\ln x ;[2,4] ; n=20$
Use a calculator to compute the left sum, right sum, and midpoint sum for the given function and interval, using a partition having the indicated number of subintervals of the same length.$f(x)=e^{2 x} ;[0,3] ; n=100$
In Exercises $25-27$ approximate the area $A$ of the region between the graph of $f$ and the $x$ axis on the given interval by using the indicated Riemann sum and a partition having the indicated number of subintervals of the same length.$f(x)=1 / x^{2} ;[1,3] ;$ left sum; $n=4$
Approximate the area $A$ of the region between the graph of $f$ and the $x$ axis on the given interval by using the indicated Riemann sum and a partition having the indicated number of subintervals of the same length.$f(x)=|x| ;[-2,2] ;$ midpoint sum; $n=8$
Approximate the area $A$ of the region between the graph of $f$ and the $x$ axis on the given interval by using the indicated Riemann sum and a partition having the indicated number of subintervals of the same length.$f(x)=\tan x ;[0, \pi / 3] ;$ upper sum; $n=50$
Why is it impossible to find a function $f$ and a partition $P$ such that $L_{f}(P)=3$ and $U_{f}(P)=2 ?$
Let $P$ be a partition of $[0,1]$, and let$$f(x)=\left\{\begin{array}{ll}1 & \text { for } x=0 \\\frac{1}{x} & \text { for } 0<x \leq 1\end{array}\right.$$Try to find $U_{f}(P)$. What difficulty do you encounter? Does the same problem arise in trying to find $L_{f}(P) ?$
Assume that $f$ and $g$ are continuous and nonnegative on $[a, b]$ and that $f(x) \leq g(x)$ for $a \leq x \leq b$. Show that for any partition $P$ of $[a, b]$ the inequalities$$L_{f}(P) \leq L_{g}(P) \quad \text { and } \quad U_{f}(P) \leq U_{g}(P)$$hold.
Suppose that $f$ is increasing on $[a, b]$ and $P=$ $\left\{x_{0}, x_{1}, x_{2}, \ldots, x_{n}\right\}$ is a partition of $[a, b]$ such that$$\Delta x_{k}=\frac{b-a}{n} \text { for } 1 \leq k \leq n$$Show that$$U_{f}(P)-L_{f}(P)=[f(b)-f(a)]\left(\frac{b-a}{n}\right)$$
Suppose $f$ is continuous on $[1,3]$ and has values appearing in the following table:$$\begin{array}{|c|c|c|c|c|c|c|}\hline x & 1 & 1.5 & 1.7 & 2.1 & 2.5 & 3 \\\hline f(x) & 2 & 1 & .5 & .2 & 0 & .1 \\\hline\end{array}$$Approximate the area $A$ between the graph of $f$ and the $x$ axis on $[1,3]$ by using the information given in the tableand an appropriatea. left sumb. right sum
Suppose a campus bookstore projects that the profits (in thousands of dollars) per month for the 4 -month period from September 1 to December 31 will be given by the function$$\bar{P}(x)=\frac{10^{4} x}{1+x^{2}} \quad \text { for } 0 \leq x \leq 4$$a. Using the lower sum corresponding to monthly increments in time, approximate the projected profit.b. Using the lower sum corresponding to semimonthly increments in time, approximate the projected profit. Explain why your answer is larger than the answer to part (a).
A skydiver drops from an airplane. At the end of each of the first six seconds the diver's speed (in meters per second) is checked, and reads as follows:$$\begin{array}{|c|c|c|c|c|c|c|}\hline \text { time } & 1 & 2 & 3 & 4 & 5 & 6 \\\hline \text { speed } & 20 & 37 & 45 & 50 & 53 & 55 \\\hline\end{array}$$Use appropriate left and right sums to approximate the distance the diver falls during the six-second period. Howmuch do they differ by?