Definite Integrals

For each function f and interval [a, b] in Exercises 26–37, use definite integrals and the Fundamental Theorem of Calculus to find the exact values of (a) the signed area and (b) the absolute area of the region between the graph of f and the x-axis from x = a to x = b.

$f\left(x\right)=\frac{3-2x}{x^2},[a,b]=[1,4]$

Write each expression in Exercises 41–43 in one sigma notation (with some extra terms added to or subtracted from the sum, as necessary). 

$3\sum _{k=2}^{25}{k}^2+2\underset{k=2}{\overset{24}{\sum k}}-\sum _{k=0}^{25}1$

Integral Formulas: Fill in the blanks to complete each of the following integration formulas.

$\int \cosh xdx=........$ 

Integral Formulas: Fill in the blanks to complete each of the following integration formulas.

$\int sec{\mathrm{h}}^{2}xdx=...$ 

For each pair of functions f and g and interval $[a, b]$ in Exercises 41–52, use definite integrals and the Fundamental Theorem of Calculus to find the exact area of the region between the graphs of $f$ and $g$ from $x=a \mathrm{to} x=b$

$f\left(x\right)=x^2-x-1$ and $g\left(x\right)=5-x^2$,$\left[-2,3\right]$

Integral Formulas: Fill in the blanks to complete each of the following integration formulas.

$\int \frac{1}{1-x^2}dx=...$. 

Indefinite integrals of combinations: Fill in the blanks to complete the integration rules that follow. You may assume that f and g are continuous functions and that k is any real number.

$\int kf\left(x\right)dx=...$ 

Indefinite integrals of combinations: Fill in the blanks to complete the integration rules that follow. You may assume that f and g are continuous functions and that k is any real number.

$\int \left(f\left(x\right)+g\left(x\right)\right)dx=...$ 

Indefinite integrals of combinations: Fill in the blanks to complete the integration rules that follow. You may assume that f and g are continuous functions and that k is any real number.

$\int \left(f^{\text{'}}\left(x\right)g\left(x\right)+f\left(x\right)g^{\text{'}}\left(x\right)\right)dx=...$

Approximations and limits: Describe in your own words how the slope of a tangent line can be approximated by the slope of a nearby secant line. Then describe how the derivative of a function at a point is defined as a limit of slopes of secant lines. What is the approximation/limit situation described in this section?

Properties of addition: State the associative law for addition, the commutative law for addition, and the distributive law for multiplication over addition of real numbers. (You may have to think back to a previous algebra course.)

Sum and constant-multiple rules: State the sum and constant-multiple rules for (a) derivatives and (b) limits.

Explain why it would be difficult to write the sum $\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{8}+\frac{1}{11}+\frac{1}{12}+\frac{1}{13}$ in sigma notation.

Use a sentence to describe what the notation $\sum _{k=2}^{100}\sqrt{k}$ means. (Hint: Start with “The sum of....”)

Use a sentence to describe what the notation $\sum _{k=3}^{87}{k}^2$ means. (Hint: Start with “The sum of....”) 

Consider the general sigma notation $\sum _{k=m}^n{a}_k$. What do we mean when we say that a_k is a function of k?

Consider the sum $\underset{i=p}{\overset{q}{\sum b_i}}$

(a) Write out this sum in expanded form (i.e., without sigma notation). 

(b) What is the index of the sum? What is the starting value? What is the ending value? Which part of the notation describes the form of each of the terms in the sum? 

(c) Do p and q have to be integers? Can they be negative? What about $b_i$? What else can you say about p and q?

Consider the sum $\sum _{k=2}^5\frac{k}{1-k}$. Identify the terms $a_2,a_3, a_4, a_5$

Consider the sum $\sum _{k=m}^n{a}_k=9+16+25+36+49$

Show that $\sum _{k=3}^9\frac{k}{k+1} \mathrm{is} \mathrm{equal} \mathrm{to} \sum _{k=4}^{10}\frac{k-1}{k}$by writing out the terms in each sum.

Show that $\sum _{k=0}^8\frac{1}{k^2+1} \mathrm{is} \mathrm{equal} \mathrm{to} 2\sum _{k=0}^8\frac{1}{2k^2+2}$ by writing out the terms in each sum.

Write the sum $\frac{4}{7}+\frac{5}{8}+\frac{6}{9}+\frac{7}{10}+\frac{8}{11}$in sigma notation in three ways: with a starting value of $\left(a\right)k=4, \left(b\right) k=7, \left(c\right) k=5$

Write the sum $2+\frac{2}{4}+\frac{2}{9}+\frac{2}{16}+\frac{2}{25}$in sigma notation in three ways: with a starting value of $\left(a\right)k=1, \left(b\right)k=2, \left(c\right)k=0$

Split the sum $\sum _{k=4}^{11} \sqrt{k}$ into three sums, each in sigma notation, where the first sum has two terms and the last two sums each have three terms. 

Verify that $\underset{k=1}{\overset{n}{\sum k}} \mathrm{is} \mathrm{equal} \mathrm{to} \frac{n(n+1)}{2}$for the cases $\left(a\right) n=2, \left(b\right) n=8, \left(c\right) n=9$

Verify that $\sum _{k=1}^n{k}^2 \mathrm{is} \mathrm{equal} \mathrm{to} \frac{n(n+1)(2n+1)}{6}$for the cases  $\left(a\right) n=1, \left(b\right) n=5,\left(c\right) n=10$

State algebraic formulas that express the following sums, where n is a positive integer:
$$\begin{gathered} \left(a\right) \sum _{k=1}^{n}1 \\ \left(b\right) \sum _{k=1}^{n}k \\ \left(c\right) \sum _{k=1}^n{k}^2 \\ \left(d\right) \sum _{k=1}^n{k}^3 \end{gathered}$$

Explain why terms in the sum in Example 6 with n equals to 4 are completely different from the terms in the sum when n equals to 3. How can the sum from $k=1 \mathrm{to} k=4$ be smaller than the sum from $k=1 to k=3$? What will happen as n gets larger in this example? 

Considering the discussion at the end of the stoplight example in the reading, would you expect that the area under the graph of a function f is related to the derivative f'? Or would you expect that the area under the graph of a derivative function f' is related to the function f? 

Consider again the stoplight example from the reading. In making an approximation for distance travelled, why do we assume that velocity is constant on small subintervals? What are some different ways that we could choose which velocity to use on each subinterval? Illustrate a couple of these ways with graphs that involve rectangles. 

Write each of the sums in Exercises 21–28 in sigma notation. Identify $m,n \mathrm{and} a_k$ in each problem.

$3+3+3+3+3+3+3+3$

Write each of the sums in Exercises 21–28 in sigma notation. Identify $m,n,a_k$ in each problem.

$\frac{4}{3}+\frac{5}{4}+\frac{6}{5}+\frac{7}{6}+\frac{8}{7}+\frac{9}{8}+\frac{10}{9}+\frac{11}{10}$

Write each of the sums in Exercises 21–28 in sigma notation. Identify $m,n,a_k$ in each problem. 

$3+\frac{4}{8}+\frac{5}{27}+\frac{6}{64}+\frac{7}{125}$

Write each of the sums in Exercises 21–28 in sigma notation. Identify $m,n,a_k$ in each problem.  

$\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}+\frac{1}{36}+\frac{1}{49}+\frac{1}{64}$

Write each of the sums in Exercises 21–28 in sigma notation. Identify $m,n,a_k$ in each problem.  

$5+10+17+26+37+50+65+82+101$

Write each of the sums in Exercises 21–28 in sigma notation. Identify $m,n,a_k$ in each problem.  

$9+12+15+18+21+24+27$

Write each of the sums in Exercises 27 in sigma notation. Identify $m$, $n$, and $a_k$in each problem.

$\frac{1}{n}+\frac{2}{n}+\frac{3}{n}+\cdots +\frac{n}{n}$

Write each of the sums in Exercises 28 in sigma notation. Identify $m$ , $n$, and $a_k$ in each problem.

$-2^n-1^n+0^n+1^n+\cdots +n^n$

Write out each sum in Exercises 29 in expanded form, and then calculate the value of the sum.

$\sum _{k=4}^9{k}^2$

Write out each sum in Exercises 30 in expanded form, and then calculate the value of the sum.

$\sum _{k=0}^6\frac{2}{k+1}$

Write out each sum in Exercises 31 in expanded form, and then calculate the value of the sum.

Write out each sum in Exercises 32 in expanded form, and then calculate the value of the sum.

$\sum _{k=3}^{10}\ln k$

Write out each sum in Exercises 33 in expanded form, and then calculate the value of the sum. 

$\sum _{k=0}^9\sqrt{3+\frac{1}{10}k}·\left(\frac{1}{10}\right)$

Write out each sum in Exercises 34 in expanded form, and then calculate the value of the sum.

$\sum _{k=1}^4\left((2+k{)}^2+1\right)$

Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading. 

(a) A sum that would not be suitable for expressing in sigma notation. 

(b) Two different sigma notation expressions of the same sum. 

(c) A sum from $k=1$ to $k=n$ that is smaller for $n=10$ than it is for $n=5$. 

Find a formula for each of the sums in Exercises 35, and then use these formulas to calculate each sum for $n=100,n=500$ and $n=1000$.

$\sum _{k=1}^n(3-k)$

Find a formula for each of the sums in Exercises 36, and then use these formulas to calculate each sum for $n=100,500$ and $n=1000$

$\sum _{k=1}^n\left(k^3-10k^2+2\right)$

Find a formula for each of the sums in Exercises 37, and then use these formulas to calculate each sum for $n=100,500$ and $1000$.

$\sum _{k=3}^n(k+1{)}^2$

Find a formula for each of the sums in Exercises 38, and then use these formulas to calculate each sum for $n=100,500$ and $1000$

Find a formula for each of the sums in Exercises 39, and then use these formulas to calculate each sum for $n=100,500$ and $1000$

$\sum _{k=1}^n\frac{k^3-1}{n^4}$