91Ó°ÊÓ

Although the definite integral of a sum of functions is equal to the sum of the definite integrals of those functions, the definite integral of a product of functions is not the product of two definite integrals.

1. Use mathematical notation to write the preceding sentence in this form:

_____ $=$______ , but _____ $â‰$______.

b. Choose two simple functions $f$and $g$so that you can calculate the definite integrals of $f$, $g$, and $f+g$on $\left[0,1\right]$, and show that the sum of the first two definite integrals is equal to the third.

c. Find two simple functions $f$and $g$such that ${∫}_0^{1}f\left(x\right)g\left(x\right)dx$is not equal to the product of ${∫}_0^{1}f\left(x\right)dx$and ${∫}_0^{1}g\left(x\right)dx$(Hint: Choose $f$and $g$so that you can calculate the definite integrals involved.)

Verify that$∫\ln xdx=x(\ln x-1)+C$(Do not try to solve the integral from scratch.

Let $f$be the function shown earlier at the right, and define $A\left(x\right)={∫}_0^{x}f\left(t\right)dt$. On which interval(s) is $A$positive? Negative? Increasing? Decreasing? Sketch a rough graph of $A$.

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

In the proof of the Fundamental Theorem of Calculus we encounter a telescoping sum. Find the values of the following sums, which are also telescoping.

$$\begin{gathered} \left(a\right)\underset{k=1}{\overset{100}{∑}}\left(\frac{1}{k}-\frac{1}{k+1}\right) \\ \left(b\right)\underset{k=1}{\overset{100}{∑}}\frac{k^2-{\left(k-1\right)}^2}{2} \end{gathered}$$

Calculating definite integrals with limits of Riemann sums: Calculate

the exact value of each the following definite integrals by

setting up a general Riemann sum and then taking the limit

as n→∞.

${∫}_{-2}^3\left(4-x^2\right)dx$

The algebra of sums: Fill in the blanks to complete the sum rules that follow. You may assume that $a_k$and $b_k$are functions defined for nonnegative integers $k$and that $c$is any real number.

$\underset{k=1}{\overset{n}{∑}}\left(a_k+b_k\right)=$______

Suppose f is an integrable function [a, b] and k is a

real number. Use pictures of Riemann sums to illustrate

that the right sum for the function kf (x) on [a, b] is k

times the value of the right sum (with the same n) for

f on [a, b]. What happens as n→∞? What does this

exercise say about the definite integrals

${∫}_a^{b}f\left(x\right)dxand{∫}_a^{b}kf\left(x\right)dx?$

Explain why the sum $\underset{k=1}{\overset{100}{∑}}f\left(2+0.1\left(k-1\right)\right)\left(0.1\right)$can't be a left sum for f on$\left[a,b\right]=\left[2,5\right]$

Suppose $f$is an integrable function $[a,b]$and $k$is a real number. Use pictures of Riemann sums to illustrate that the right sum for the function $kf\left(x\right)$on $[a,b]$is $k$times the value of the right sum (with the same $n$) for $f$on $[a,b]$. What happens as $n→∞$? What does this exercise say about the definite integrals${∫}_a^{b}f\left(x\right)dx$and${∫}_a^{b}kf\left(x\right)dx?$