Question

Let V be the volume of the region S ⊆ \(\R^3\) defined by
S = {(x, y, z) ∈ \(\R^3\) : xy ≤ z ≤ 4, 0 ≤ x² + y² ≤ 1}.
Then \(\frac{V}{π}\) is equal to ________ . (rounded off to two decimal places)

Answer 1

Correct Answer: 3.99 - 4.01

We need to find the volume \( V \) of the region \( S \subseteq \mathbb{R}^3 \) defined by:

1. \( xy \leq z \leq 4 \) 2. \( 0 \leq x^2 + y^2 \leq 1 \)

The first condition \( xy \leq z \leq 4 \) suggests a region bounded above by \( z = 4 \) and below by \( z = xy \). The second condition \( 0 \leq x^2 + y^2 \leq 1 \) describes a cylinder with a circular base of radius 1 in the xy-plane. To compute the volume, consider integrating over the region where \( x^2 + y^2 \leq 1 \). We switch to cylindrical coordinates (since the region is circularly symmetric in the \( x \) and \( y \) coordinates): - \( x = r \cos \theta \) - \( y = r \sin \theta \) - \( z \) varies from \( xy = r^2 \cos \theta \sin \theta \) to 4. Thus the volume integral in cylindrical coordinates is: \[ V = \int_{\theta=0}^{2\pi} \int_{r=0}^{1} \int_{z=r^2 \cos \theta \sin \theta}^{4} r \, dz \, dr \, d\theta \] Evaluating the inner integral (over \( z \)): \[ \int_{z=r^2 \cos \theta \sin \theta}^{4} r \, dz = r(4 - r^2 \cos \theta \sin \theta) \] Now, evaluate the integrals over \( r \) and \( \theta \): \[ \int_{\theta=0}^{2\pi} \int_{r=0}^{1} \left(4r - r^3 \cos \theta \sin \theta\right) \, dr \, d\theta \] Separate the terms: 1. For \( 4r \): \[ \int_{\theta=0}^{2\pi} \int_{r=0}^{1} 4r \, dr \, d\theta = \int_{\theta=0}^{2\pi} \left[2r^2\right]_{0}^{1} \, d\theta = \int_{\theta=0}^{2\pi} 2 \, d\theta = 4\pi \] 2. For \( -r^3 \cos \theta \sin \theta \): \[ \int_{\theta=0}^{2\pi} \int_{r=0}^{1} -r^3 \cos \theta \sin \theta \, dr \, d\theta \] The integral of \( \cos \theta \sin \theta \) over \( 0 \) to \( 2\pi \) is zero because it is an odd function with symmetric limits. Therefore, this term evaluates to zero. So the volume \( V = 4\pi \). Hence, \(\frac{V}{\pi} = \frac{4\pi}{\pi} = 4\). The computed value is 4, and it falls within the given range of 3.99 to 4.01.