009B Sample Final 1, Problem 5

The region bounded by the parabola  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=x^2}   and the line  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=2x}   in the first quadrant is revolved about the  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} -axis to generate a solid.

(a) Sketch the region bounded by the given functions and find their points of intersection.

(b) Set up the integral for the volume of the solid.

(c) Find the volume of the solid by computing the integral.

Foundations:
1. You can find the intersection points of two functions, say   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x),g(x),}
by setting  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=g(x)}   and solving for  $x.$
2. The volume of a solid obtained by rotating an area around the  $y$-axis using cylindrical shells is given by
$\int 2\pi rh~dx,$  where  $r$  is the radius of the shells and  $h$  is the height of the shells.

Solution:

(a)

Step 1:
First, we sketch the region bounded by the given functions.
300px

Step 2:
Setting the equations equal, we have  $x^{2}=2x.$
Solving for  $x,$  we get
${\begin{array}{rcl}\displaystyle {0}&=&\displaystyle {x^{2}-2x}\\&&\\&=&\displaystyle {x(x-2).}\end{array}}$
So,  $x=0$  and  $x=2.$
If we plug these values into our functions, we get the intersection points
$(0,0)$  and  $(2,4).$
This intersection points can be seen in the graph shown in Step 1.

(b)

Step 1:
We proceed using cylindrical shells. The radius of the shells is given by  $r=x.$
The height of the shells is given by  $h=2x-x^{2}.$

Step 2:
So, the volume of the solid is
$\int 2\pi rh~dx~=~\int _{0}^{2}2\pi x(2x-x^{2})~dx.$

(c)

Step 1:
We need to integrate
$\int _{0}^{2}2\pi x(2x-x^{2})~dx~=~2\pi \int _{0}^{2}2x^{2}-x^{3}~dx.$

Step 2:
We have
${\begin{array}{rcl}\displaystyle {\int _{0}^{2}2\pi x(2x-x^{2})~dx}&=&\displaystyle {2\pi \int _{0}^{2}2x^{2}-x^{3}~dx}\\&&\\&=&\displaystyle {2\pi {\bigg (}{\frac {2x^{3}}{3}}-{\frac {x^{4}}{4}}{\bigg )}{\bigg |}_{0}^{2}}\\&&\\&=&\displaystyle {2\pi {\bigg (}{\frac {2^{4}}{3}}-{\frac {2^{4}}{4}}{\bigg )}-2\pi (0)}\\&&\\&=&\displaystyle {{\frac {8\pi }{3}}.}\\\end{array}}$

Final Answer:
(a)    $(0,0),(2,4)$  (See Step 1 for the graph)
(b)    $\int _{0}^{2}2\pi x(2x-x^{2})~dx$
(c)    ${\frac {8\pi }{3}}$

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