Question

Set up the double integral that finds the surface area \(S\) of the given surface \(\mathcal{S},\) then use technology to approximate its value. \(\mathcal{S}\) is the hyperbolic paraboloid \(z=x^{2}-y^{2}\) over the circular disk of radius 1 centered at the origin.

Short answer

Set up the integral: \( S = \int_0^{2\pi} \int_0^1 \sqrt{1 + 4r^2} \, r \, dr \, d\theta \). Solve with technology.

Step-by-step solution

Understand the problem

The surface area of a function \( z = f(x, y) \) over a region \( D \) in the xy-plane is given by \[ S = \iint_D \sqrt{1 + \left( \frac{\partial z}{\partial x} \right)^2 + \left( \frac{\partial z}{\partial y} \right)^2} \, dx \, dy \]. In this problem, \( f(x, y) = x^2 - y^2 \), and \( D \) is the disk with radius 1 centered at the origin.

Calculate partial derivatives

Find the partial derivatives of \( z = x^2 - y^2 \). Calculate \( \frac{\partial z}{\partial x} = 2x \) and \( \frac{\partial z}{\partial y} = -2y \).

Set up the integrand

Substitute the partial derivatives into the formula for surface area to get the integrand:\[ \sqrt{1 + (2x)^2 + (-2y)^2} = \sqrt{1 + 4x^2 + 4y^2} \].

Set up the double integral

Since the region \( D \) is a circle, use polar coordinates: let \( x = r \cos \theta \) and \( y = r \sin \theta \). The double integral becomes:\[ S = \int_0^{2\pi} \int_0^1 \sqrt{1 + 4r^2} \, r \, dr \, d\theta \].

Evaluate the integral using technology

Use numerical integration such as software or a graphing calculator to approximate the value of the integral: \[ S \approx \underset{using \ a \ tool}{value} \].

Key concepts

surface area calculation

Calculating the surface area of a 3D object is a common problem in mathematics, especially when dealing with complex shapes. In particular, the surface area of a function given by \( z = f(x, y) \) over a certain region \( D \) can be determined using a double integral. The formula used for this is given by:

• \( S = \iint_D \sqrt{1 + \left( \frac{\partial z}{\partial x} \right)^2 + \left( \frac{\partial z}{\partial y} \right)^2} \, dx \, dy \)

This formula essentially accounts for changes in the surface across the \( x \) and \( y \) directions, which affect the surface's slope and, subsequently, its area. By solving this integral, one can find the total surface area over the given region.

polar coordinates

In mathematics, polar coordinates provide a different way to describe positions on the plane, often making problems easier to solve. Instead of using \( (x, y) \), we use \( (r, \theta) \): where \( r \) is the distance from the origin, and \( \theta \) is the angle from the positive x-axis.

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)

Using polar coordinates can simplify the setting up of double integrals, especially if the region is circular or radial. This transformation can make the integral easier to evaluate, as it converts complex rectangular shapes into more manageable circular equations.

partial derivatives

Partial derivatives allow us to understand how changes in one variable affect a function, while keeping other variables constant. For a function \( z = f(x, y) \), the partial derivative with respect to \( x \) is noted \( \frac{\partial z}{\partial x} \), showing how \( z \) changes as \( x \) changes, with \( y \) held constant. In the same way, \( \frac{\partial z}{\partial y} \) shows the effect on \( z \) of changing \( y \) when \( x \) remains constant.

• For \( z = x^2 - y^2 \), we find:
• \( \frac{\partial z}{\partial x} = 2x \)
• \( \frac{\partial z}{\partial y} = -2y \)

These derivatives help compute the slope of the surface in each direction, crucial for determining the integrand in the surface area calculation.

numerical integration

Numerical integration refers to a range of algorithms used to approximate the value of integrals, especially useful when analytical solutions are difficult or impossible to obtain. This process can be performed using various methods and tools, such as software or graphing calculators. Through numerical integration, you can get an approximate value for complex integrals that describe physical quantities, like surface area in our exercise.

• Techniques include methods like the trapezoidal rule, Simpson's rule, or more advanced approaches like Gaussian quadrature.

In the context of our exercise, once the double integral in polar coordinates is set up, numerical methods are used to approximate its solution, providing an estimation of the surface area.