Question

Let f (x) = 9x and g(x) = x2 + 6x, and let r be the bounded region between the graphs of f and g. set up an integral to find the volume v of the solid generated by revolving r about the x axis and then find v.

Answer 1

find where they intersect
f(x)=g(x) at

ok
9x=x^2+6x
0=x^2-3x
intersect at x=0 and x=3
which one is on 'top'
f(2)=18
g(2)=16
f(x) is on top

so it is
(we don't need to find r)
remember, the volume rotated around  x axis when f(x) is top curve and g(x) is botttom and bounds are a and b is
$\int\limits^a_b {(f(x))^2-(g(x))^2} \, dx$
so
f(x)=9x
g(x)=x^2+6x
the bounds are 0 and 3
$\int\limits^3_0 {(9x)^2-(x^2+6x)^2} \, dx$=v
$\int\limits^3_0 {x^4+12x^3+117x^2} \, dx$=v
$[ \frac{x^5}{5}+3x^4+ 39x^3 ]^3_0$=v
$\frac{243}{5}+1296=v$
the volume is 1344.6 cubic units

Answer 2

The volume of the solid generated by revolving about the x-axis is $\frac{567}{5}\pi$ cubic units.

How to generate to a solid of revolution of a region around the x-axis

The points of intersection between $f(x)$ and $g(x)$ are $(0, 0)$ and $(3, 27)$. The formula of a solid of revolution generated around the x-axis is represented by the following integral formula:

$V = \pi \int\limits^{b}_{a} {\{[f(x)]^{2}-[g(x)]^{2}\}} \, dx$   (1)

If we know that $a = 0$, $b = 3$, $f(x) = 9\cdot x$ and $g(x) = x^{2}+6\cdot x$, then the volumen of the solid about the x-axis is:

$V = \pi \int\limits^{3}_{0} {[9\cdot x]^{2}} \, dx -\pi \int\limits^{3}_{0} {[x^{2}+6\cdot x]^{2}} \, dx$

$V = 81\pi \int\limits^3_0 {x^{2}} \, dx - \pi \int\limits^{3}_{0} {[x^{4}+12\cdot x^{3}+36\cdot x^{2}]} \, dx$

$V = -\pi \int\limits^{3}_{0} {x^{4}} \, dx - 12\pi \int\limits^{3}_{0} {x^{3}} \, dx +45\pi\int\limits^{3}_{0} {x^{2}} \, dx$

$V = -\frac{\pi}{5}\cdot (3^{5}-0^{5}) - \frac{12\pi}{4}\cdot (3^{4}-0^{4}) +\frac{45\pi}{3}\cdot (3^{3}-0^{3})$

$V =\frac{567}{5}\pi$

The volume of the solid generated by revolving about the x-axis is $\frac{567}{5}\pi$ cubic units. $\blacksquare$

To learn more on solids of revolution, we kindly invite to check this verified question: https://brainly.com/question/338504