Find the volume of the solid lying in side both the sphere bounded x^2+y^2+z^2=4a^2 and the cylinder x^2+y^2=2ay
To find the volume of the solid lying inside both the sphere and the cylinder, we need to determine their intersection. Let's first find the intersection curve between the sphere and the cylinder.
For both equations:
x^2 + y^2 + z^2 = 4a^2
x^2 + y^2 = 2ay
We can substitute x^2 + y^2 from the cylinder equation into the sphere equation:
2ay + z^2 = 4a^2
Now, we can isolate z^2:
z^2 = 4a^2 - 2ay
To find the intersection curve, we need to solve for z in terms of x and y. We can take the square root of both sides:
z = ±√(4a^2 - 2ay)
Now that we have the equation of the intersection curve, let's determine the limits of integration for the volume calculation.
We'll integrate over the x-y plane, so the limits for both x and y are determined by the cylinder equation:
x^2 + y^2 = 2ay
Let's solve this equation for y:
y = (x^2)/(2a)
The limits for x will be determined by the intersection points between the cylinder and the sphere. To find these points, we can set z = 0 in the equation for the intersection curve:
0 = ±√(4a^2 - 2ay)
Simplifying, we get:
4a^2 - 2ay = 0
2ay = 4a^2
y = 2a
Now let's substitute the expression for y into the equation of the cylinder to find the limits for x:
x^2 + (2a)^2 = 2a(2a)
x^2 + 4a^2 = 4a^2
x^2 = 0
x = 0
Therefore, the limits for x and y are from -√(2a) to √(2a), and 0 to 2a, respectively.
Now, we can calculate the volume of the solid using triple integration. Since the solid lies inside both the sphere and the cylinder, the volume integral is given by:
V = ∬∬∬ dV
Where dV is the infinitesimal volume element.
Integrating over the sphere and cylinder equations, we obtain:
V = ∫[∫[∫[ 1 dz ] dy ] dx ]
The limits of integration will be as follows:
x: -√(2a) to √(2a)
y: 0 to 2a
z: -√(4a^2 - 2ay) to √(4a^2 - 2ay)
Evaluating this triple integral will give us the volume of the solid lying inside both the sphere and the cylinder.