Evaluate the triple integral ∫∫∫_E (x)dV
where E is the solid bounded by the paraboloid x=10y^2+10z^2 and x=10
Introduce polar coordinates in the y-z plane. You can then write the integral as:
Integral over theta from 0 to 2 pi
Integral over r from 0 to 1
Integral over x from 10 r^2 to 10
r dr dtheta dx
100/3*pi
To evaluate the given triple integral, we need to first describe the solid E in terms of its boundaries.
The paraboloid is defined as x = 10y^2 + 10z^2, and it is bounded by x = 10.
We can rewrite the equation of the paraboloid as:
x = 10(y^2 + z^2)
Now, let's express the domain of integration in terms of cylindrical coordinates, where:
x = ρcos(θ)
y = ρsin(θ)
z = z
To determine the limits of integration, we need to find the range of ρ, θ, and z that define the solid E.
From the equation x = 10(y^2 + z^2) and x = 10, we can set these equal to each other:
10(y^2 + z^2) = 10
Simplifying, we get:
y^2 + z^2 = 1
This represents the equation of a cylinder of radius 1 centered at the origin in the yz-plane.
Considering the given boundary x = 10, we can rewrite it in cylindrical coordinates:
10 = ρcos(θ)
Solving for ρ, we get:
ρ = 10/cos(θ)
Now we have the limits of integration as:
ρ: from 0 to 10/cos(θ)
θ: from 0 to 2π
z: from -√(1 - ρ^2) to √(1 - ρ^2)
Finally, we can set up the triple integral as follows:
∫∫∫_E (x)dV = ∫(θ=0 to 2π) ∫(ρ=0 to 10/cos(θ)) ∫(z=-√(1 - ρ^2) to √(1 - ρ^2)) (x) (ρ dz dρ dθ)
Now, you can proceed to evaluate this triple integral using the given limits of integration.
To evaluate the triple integral ∫∫∫_E (x)dV, where E is the solid bounded by the paraboloid x=10y^2+10z^2 and x=10, we will use cylindrical coordinates.
Step 1: Sketch the region of integration.
The region of integration is bounded by the paraboloid x=10y^2+10z^2 and the plane x=10. This represents a solid that lies inside a cylinder of radius 10 and height 10 in the x-direction.
Step 2: Determine the boundaries for each variable in cylindrical coordinates.
Since we are using cylindrical coordinates, we need to express x, y, and z in terms of cylindrical variables: r, θ, and z.
From the equation x=10, it is clear that r=10.
From the equation x=10y^2+10z^2, we can rewrite it as r=10y^2+10z^2.
Since we are given the limits for x, we have r ranging from 0 to 10.
For θ, it ranges from 0 to 2π since we want to cover the full circle.
For z, it ranges from 0 to the value of z at the surface of the paraboloid, which can be found by setting x=10 and solving for z using the equation x=10y^2+10z^2.
Step 3: Set up and evaluate the integral.
The triple integral can now be written as:
∫∫∫_E (x)dV = ∫[0 to 2π]∫[0 to 10]∫[0 to z] (r*cosθ)r dz dr dθ
It is important to note that (x)dV is equivalent to (r*cosθ)(r dz dr dθ) in cylindrical coordinates, where r represents the magnitude of a position vector in the xy-plane and cosθ represents the direction of the position vector in the x-direction.
Evaluating this triple integral will involve integrating with respect to z, r, and θ in that order.