How do you find the volume of region bounded by graphs of #y = x^2# and #y = sqrt x# about the x-axis?

Question

Ezra Adams · Accepted Answer

#color(blue)(pi/3 "cubic units.")#

From the graph we can see that the volume we seek is between the two functions. In order to find this, we must find the volume of revolution of #f(x)=sqrt(x)# and subtract the volume of revolution of #f(x)=x^2#. This is shown as the shaded area.

First we need to find the upper and lower bounds. We know the lower bound is #0# since #f(x)=sqrt(x)# is undefined for #x<0#. The upper bound is where the functions intersect:

#:.#

#x^2=sqrt(x)#

#x^2/x^(1/2)=1#

#x^(3/2)=1#

Squaring:

#x^3=1#

#x=root(3)(1)=1#

Volume of #bb(f(x)=sqrt(x))#:

#pi int_(0)^(1)(x^(1/2))=pi[2/3x^(3/2)]_(0)^(1)#

#=pi{[2/3x^(3/2)]^(1)-[2/3x^(3/2)]_(0)}#

Plugging in upper and lower bounds:

#=pi{[2/3(1)^(3/2)]^(1)-[2/3(0)^(3/2)]_(0)}=(2pi)/3# cubic units

Volume of #bb(f(x)=x^2)#

#pi int_(0)^(1)(x^2)=pi[1/3x^3]_(0)^(1)#

#=pi{[[1/3x^3]^(1)-[1/3x^3]_(0)}#

Plugging in upper and lower bounds:

#=pi{[[1/3(1)^3]^(1)-[1/3(0)^3]_(0)}=pi/3# cubic units.

Required volume is:

#(2pi)/3-pi/3=##color(blue)(pi/3 "cubic units.")#

Volume of revolution:

Emma Aldridge · Answer

undefined To find the volume of the region bounded by the graphs of $y = x^2$ and $y = \sqrt{x}$ about the x-axis, you would integrate the difference between the two functions squared, from the x-coordinate of their intersection to the x-coordinate where they end or begin. The integral formula is:

$$V = \pi \int_{a}^{b} \left((f(x))^2 - (g(x))^2\right) \, dx$$

where:
$f(x)$ is the upper function,
$g(x)$ is the lower function,
$a$ and $b$ are the x-coordinates of the intersection points.

In this case:
$f(x) = x^2$ (upper function)
$g(x) = \sqrt{x}$ (lower function)

First, you need to find the intersection points by setting the two equations equal to each other and solving for $x$. Then, integrate the difference between $f(x)$ and $g(x)$ squared from the lower x-coordinate to the upper x-coordinate of the intersection points. Finally, multiply the result by $\pi$ to get the volume.