Answer:
The value is [tex]V(x) = 1706.7 \pi[/tex]
Step-by-step explanation:
From the question we are told that
The first equation is [tex]y=16\sqrt{x}[/tex]
The second equation is [tex]y=4x[/tex]
Generally the first point of intersection of the first and second equation is x = 0
Generally the obtain the second point of intersection of the two equation we equate the two equations
So
=> [tex]16\sqrt{x} = 4x[/tex]
=> [tex]\sqrt{x} = \frac{4x}{16}[/tex]
=> [tex]x = 4[/tex]
Generally the from washer method we have
[tex]V(x) = \int\limits^4_0 {\pi [(H(x))^2 - (G(x))^2]} \, dx[/tex]
So
[tex]H(x) = 16\sqrt{x}[/tex]
and
[tex]G(x) = 4x[/tex]
So
[tex]V(x) = \int\limits^4_0 {\pi [(16\sqrt{x})^2 - (4x)^2]} \, dx[/tex]
=> [tex]V(x) = \int\limits^4_0 {\pi [256x - 16x^2]} \, dx[/tex]
=>[tex]V(x) = \pi [256 \frac{x^2}{2} - 16 \frac{x^3}{3} ]|\left 4} \atop 0}} \right.[/tex]
=> [tex]V(x) = \pi [256 * \frac{ 4^2}{2} - 16 * \frac{4^3}{3} ][/tex]
=> [tex]V(x) = 1706.7 \pi[/tex]
