91Ó°ÊÓ

Consider the sequence A(1), A(2), A(3),.....,A(n) write our the sequence up to n. What do you notice?

Suppose $f\left(x\right)â‰\yen g\left(x\right)$on [1, 3] and $f\left(x\right)≤g\left(x\right)$on (−∞, 1] and [3,∞). Write the area of the region between the graphs of f and g on [−2, 5] in terms of definite integrals without using absolute values .

Explain geometrically what the definition of the definite integral as a limit of Riemann sums represents. Include a labeled picture of a Riemann sum (for a particular n) that illustrates the roles of n, x, $∆x$, ${x_k}^{*}$and $f\left({x^{*}}_k\right)$. What happens in the picture as n → ∞?

Prove Theorem 4.13(b): For any real numbers a and b, we have${∫}_a^{b}xdx=\frac{1}{2}\left(b^2-a^2\right)$. Use the proof of Theorem 4.13(a) as a guide.

Given a simple proof that if n is a positive integer and c is any real number, then$\underset{k=1}{\overset{n}{∑}}c=cn$

For each function f and interval[a, b] in Exercises 56–67, use definite integrals and the Fundamental Theorem of Calculus to find the exact average value of f from x = a to x = b. Then use a graph of f to verify that your answer is reasonable.

$f\left(x\right)={(x+2)}^2−5,[−5,0]$

Use integration formulas to solve each integral in Exercises 21–62. You may have to use algebra, educated guess- and- check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating. (Hint for Exercise 54: $\tan \left(x\right)=\frac{\sin \left(x\right)}{\cos \left(x\right)}$).

$∫\frac{1}{2\sqrt{x^2-1}}dx.$

Use the Fundamental Theorem of Calculus to find the exact

values of each of the definite integrals in Exercises$19–64$. Use

a graph to check your answer.

${∫}_{-1}^3\left|4-x^2\right|dx$

Jimmy is doing some arithmetic problems. As the evening wears on, he gets less and less effective, so that $tâ‰\yen 1$minutes after the start of his study session he can solve $r\left(t\right)=1+\frac{1}{t}$problems per minute. Assume that it takes $1$minute for Jimmy to set up his work area; thus he is not doing arithmetic problems in the first minute.

Part (a): How many arithmetic problems per minute can Jimmy do when he first begins his studies at time $t=1$? What about at $$\begin{gathered} t=4, \\ t=20 \end{gathered}$$?

Part (b): Make a rough estimate of the number of arithmetic problems Jimmy will have completed after $4$minutes.

Part (c): Use an integral to express the number of arithmetic problems completed after t minutes, and interpret this definite integral as a logarithm. Calculate the number of problems Jimmy can complete in $10$minutes and in $20$minutes.

Part (d): Approximately how long will it take for Jimmy to finish his arithmetic homework if he must complete $40$problems?

Prove Theorem 4.13(c): For any real numbers a and b, ${∫}_a^b{x}^{2}dx=\frac{1}{3}\left(b^3-a^3\right).$Use the proof of Theorem 4.13(a) as a guide.