Final answer:
To find the volume of the region E between a paraboloid and a cone, we solve the triple integral with cylindrical coordinates after finding the intersection of the two surfaces, which provides the limits for r.
Step-by-step explanation:
The student is asking to find the volume of the region E that is enclosed between a paraboloid represented by the equation z = 24 - x^2 - y^2 and a cone given by the equation z = 2 √(x^2 + y^2). To find this volume, we can set up an integral in cylindrical coordinates, as both surfaces are symmetrical about the z-axis. The intersection of these surfaces will determine our limits of integration.
Firstly, we find the intersection by setting the two equations equal to each other:
24 - x^2 - y^2 = 2 √(x^2 + y^2)
Squaring both sides, we get:
576 - 48x^2 - 48y^2 + x^4 + 2x^2y^2 + y^4 = 4(x^2 + y^2)
This simplifies to:
x^4 + 2x^2y^2 + y^4 - 52x^2 - 52y^2 + 576 = 0
Converting to cylindrical coordinates where x = r cos(θ) and y = r sin(θ), we get r = 4 after simplification. This radius will serve as our outer limit for the volume integration.
Next, we set up the integral. The volume V is given by:
V = \( \int_0^{2\pi} \int_0^4 \int_{2r}^{24-r^2} r dz dr d\theta \)
We integrate with respect to z first, from the lower surface (cone) to the upper surface (paraboloid), then with respect to r, and finally with respect to θ. Solving this integral will give us the desired volume of the region E.