Question

Evaluate each of the integrals as either a volume integral or a surface integral, whichever is easier. \(\iint \mathbf{r} \cdot \mathbf{n} d \sigma\) over the entire surface of the cone with base \(x^{2}+y^{2} \leq 16, z=0,\) and vertex at \((0,0,3),\) where \(\mathbf{r}=\mathbf{i} x+\mathbf{j} y+\mathbf{k} z\).

Short answer

The integral evaluates to \( 48\pi \).

Step-by-step solution

Interpret the Problem

We need to evaluate the surface integral \[ \iint_S \mathbf{r} \cdot \mathbf{n} \, d\sigma \] over the surface of a cone with base \( x^2 + y^2 \leq 16, z=0 \) and vertex at \((0,0,3)\). The position vector is given by \(\mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}\).

Simplify using Divergence Theorem

According to the Divergence Theorem:\[ \iint_S \mathbf{F} \cdot \mathbf{n} \, d\sigma = \iiint_V abla \cdot \mathbf{F} \, dV \] Set \( \mathbf{F} = \mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \), and find the divergence \( abla \cdot \mathbf{r} \).

Calculate the Divergence

Calculate \( abla \cdot \mathbf{r} \):\[ abla \cdot \mathbf{r} = \frac{\partial x}{\partial x} + \frac{\partial y}{\partial y} + \frac{\partial z}{\partial z} = 1 + 1 + 1 = 3 \]

Set Up the Volume Integral

Using the result from the previous step, the integral becomes:\[ \iiint_V 3 \, dV \].The volume of integration \(V\) is the cone with height 3 and base radius 4.

Calculate the Volume of the Cone

The volume \(V\) of the cone is given by the formula:\[ V = \frac{1}{3}\pi r^2 h \]Substitute \(r=4\) and \(h=3\):\[ V = \frac{1}{3}\pi (4)^2 (3) = \frac{1}{3} \pi (16)(3) = 16\pi \]

Evaluate the Volume Integral

Now multiply the divergence by the volume:\[ \iiint_V 3 \, dV = 3 \times 16\pi = 48\pi \]

Key concepts

vector calculus

Vector calculus is a branch of mathematics focused on vector fields and operations on them. It extends calculus concepts to functions of several variables. Basic operations include:

• Gradient (\(abla f\)), the vector of partial derivatives of a scalar function \(f\).
• Divergence (\(abla \, \cdot \mathbf{F}\)), which measures a vector field's tendency to originate from or converge into a point.
• Curl (\(abla \, \times \mathbf{F}\)), which measures the rotation of a vector field.

These operations help analyze physical fields such as fluid flow and electromagnetism. In this exercise, we utilize the Divergence Theorem, linking flux through a closed surface to volume integral of divergence inside.

surface integrals

Surface integrals calculate the flow of a vector field across a surface. They extend scalar integrals to vector fields. For a surface \(S\) with a normal vector \(\mathbf{n}\), and vector field \(\mathbf{F}\), the surface integral is:
\(\iint_S \mathbf{F} \cdot \mathbf{n} \, d\sigma \).
Here, \(\mathbf{n}\) is a unit vector perpendicular to the surface. Surface integrals have practical applications in physics and engineering:

• Determining flow rates of fluids across surfaces.
• Calculating flux in electromagnetic fields.

In the given problem, the surface integral over the cone simplifies using Divergence Theorem.

volume integrals

Volume integrals extend surface integrals into three dimensions, calculating the sum of values within a volume. They are written as: \(\iiint_V f \, dV \), where \(V\) is the volume over which the integration occurs, and \(f\) is the function to integrate.
Volume integrals are essential in:

• Determining the total mass of a 3D body from a density function.
• Calculating total energy stored in a given volume.

In this exercise, using Divergence Theorem, we convert the surface integral to a simpler volume integral, \(\iiint_V 3 \, dV\), over the volume of the cone.

cone geometry

The geometry of the cone in the problem has a circular base and a pointed vertex. The cone's dimensions are:

• Base radius, \( r = 4 \).
• Height, \( h = 3 \).

The volume \(V\) of a cone is given by: \( V = \frac{1}{3}\pi r^2 h \)
In the exercise, we calculate the volume:
\( V = \frac{1}{3}\pi (4)^2 (3) = 16\pi \)
This volume calculation is crucial, as it allows us to evaluate the volume integral derived from the Divergence Theorem, leading to: \( 3 \times 16\pi = 48\pi \). Knowledge of cone geometry helps solve integrals involving these shapes efficiently.