Respuesta :
Answer:
Option b) [tex]18\pi[/tex] is correct
∴ the volume of the solid is [tex]A(x)=18\pi[/tex] cubic units
Step-by-step explanation:
Given that the base of a solid is the circle [tex]x^2 + y^2 = 9[/tex] and Cross sections of the solid perpendicular to the x-axis are semi-circles.
To find the the volume of the solid in cubic units:
We know that the cross sections are semicircles with the diameter in the given circle [tex]x^2 + y^2 = 9[/tex]
That is we have to find the formula for the area of any semicircle perpendicular to x-axis, and integrate it from -3 to 3.
Now the area of a semicircle is
[tex]A=\frac{\pi r^2}{2}[/tex] cubic units
Let r = y and [tex]y^2=9-x^2[/tex]
Then area of the semicircle crossing the x-axis at x is given by
[tex]A(x)=\frac{1}{2}\pi y^2[/tex] cubic units
[tex]=\frac{1}{2}\pi(9-x^2)[/tex]
Now we can find the definite integral of A(x) from x = -3 to x = 3.
Since A(x) is an EVEN function then the definite integral of A(x) from x = -3 to x = 3 is the same as twice the integral of A(x) from x = 0 to x = 3.
We have that
[tex]A(x)=2(\int_0^3 \frac{1}{2}\pi(9-x^2))dx[/tex]
[tex]=2(\frac{\pi}{2}[9x-\frac{x^3}{3}]_0^3)[/tex]
[tex]=\pi[9(3)-\frac{3^3}{3}-9(0)-(-\frac{0^3}{3})][/tex]
[tex]=\pi[27-\frac{27}{3}][/tex]
[tex]=\pi[27-9][/tex]
[tex]=\pi[18][/tex]
[tex]=18\pi[/tex]
