Consider the functions

$f(x)=x^3-6x^2+9x,$ and $,g(x)=x$

graphed below.

$a.$ Set $up$ the integrals for the area between $f$ and $g$ from $0tob(thisis$ the green$-$ and yellow$-$shaded

regions$).$

Note: You will need $to$ find the $x-$values, a and $b.$

Area $={\displaystyle \underset{0}{\overset{\phantom{}}{}}}|,?o++{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}$

$b.$ Set $upan$ integral for the volume $of$ the solid obtained $by$ rotating the green$-$shaded region $(the$ first

region$)$ about the line $y=-6.$

Volume $={\displaystyle \underset{0}{\overset{\phantom{}}{}}}$

$c.$ Set $upan$ integral for the volume $of$ the solid obtained $by$ rotating the yellow$-$shaded region $(the$

second region$)$ about the line $x=5.$

Volume $={\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}b.$ Set $upan$ integral tor the volume $of$ the solid obtained $by$ rotating the green$-$shaded region $(the$ first region$)$ about the line $y=-6.$

Volume$\frac{}{b}=ar({\displaystyle \underset{0}{\overset{\phantom{}}{}}})$

$c.$ Set $upan$ integral for the volume $of$ the solid obtained $by$ rotating the yellow$-$shaded region $(the$ second region$)$ about the line $x=5.$

Volume$\frac{}{b}=ar(l),?$

$d.$ Set $upan$ integral for the arc length $off(x)$ from $x=0tox=a.$

Note: This $is$ the same $f(x)$ and a from the nrevious three narts.

Arc Length $={\displaystyle \underset{0}{\overset{\phantom{}}{}}}$

$$

$?$

$e.$ Finally, set $upan$ integral for the surface area $of$ the solid obtained $by$ rotating the shaded region $in$ part $d$ from $x=0tox=$a about the $x-$axis.

$$

Surface Area $={\displaystyle \underset{0}{\overset{\phantom{}}{}}}$

$?$