Question

Find the area of the part of the sphere of radius \(a\) and center at the origin which is above the square in the \((x, y)\) plane bounded by \(x=\pm a / \sqrt{2}\) and \(y=\pm a / \sqrt{2} .\) Hint for evaluating the integral: Change to polar coordinates and evaluate the \(r\) integral first.

Short answer

Following the steps, we use the hint to change to polar coordinates and integrate over the appropriate bounds.

Step-by-step solution

- Understand the problem

Analyze the problem of finding the area of a part of the sphere with radius \(a\) and center at the origin. The part of the sphere lies above a square in the \(xy\)-plane bounded by \(x = \pm \frac{a}{\sqrt{2}}\) and \(y = \pm \frac{a}{\sqrt{2}}\).

- Setting up the integral

To solve the problem, we will use surface area integration in spherical coordinates. The surface area element in spherical coordinates is given by \(dS = a \sin(\theta)d\theta d\phi\).

- Converting to polar coordinates

Use the hint to change the cartesian coordinates in the \(x, y\)-plane to polar coordinates. The bounds for the given square are:

Key concepts

spherical coordinates

Spherical coordinates are a system of curvilinear coordinates that are natural for describing positions on a sphere. Instead of using \(x, y, z\) Cartesian coordinates, spherical coordinates use three values: radius \(r\), polar angle \(\theta\) (also known as the colatitude), and azimuthal angle \(\theta\) (also referred to as the longitude).
The radius is the distance from the origin to the point, the polar angle measures the angle from the positive \(z\)-axis, and the azimuthal angle measures the angle in the \(xy\)-plane from the positive \(x\)-axis.

Using these coordinates, the surface element for a sphere of radius \(a\) is given by:
\[ dS = a^2 \sin(\theta) \ d\theta \ d\phi \]
This equation accounts for the curvature of the sphere when integrating over its surface.

To visualize spherical coordinates, imagine drawing a line from the center of the sphere to any point on its surface. The length of this line is the radius. The angle between this line and the positive \(z\)-axis is the polar angle \(\theta\). Finally, the angle between the projection of this line onto the \(xy\)-plane and the positive \(x\)-axis is the azimuthal angle \(\theta\).

polar coordinates

Polar coordinates are a two-dimensional coordinate system where each point in the plane is determined by a distance from a reference point and an angle from a reference direction. These coordinates are particularly useful in problems involving circular symmetry.
The reference point is called the pole (analogous to the origin in Cartesian coordinates), and the reference direction is typically the positive \(x\)-axis.

In polar coordinates, any point is identified by \(r \) and \(\theta \). The value \(|r| \) is the distance from the pole, and \(\theta \) is the angle from the positive \(x \)-axis.
To convert Cartesian coordinates \( (x, y ) \) to polar coordinates, we use:

• \[ r = \sqrt{x^2 + y^2} \ ]
• \[\theta = \text{arctan} \left(\frac{y}{x}\right) \]

Using polar coordinates simplifies integration over circular regions. When converting the given exercise's square region to polar coordinates, the bounds for \(\theta \) range from \[ \theta = -\frac{\pi}{4} \ to \ \frac{\theta}{4} \], and for \( r \) range from \ 0 \ to \ \frac{a}{\root{2}}\]

surface area element

The surface area element is an infinitesimal piece of surface area used in surface integrals. In traditional Cartesian coordinates, the surface area element is expressed differently based on the surface. For example, on a plane, it might be \( dA = dx \ dy \).

In spherical coordinates, which are suitable for surfaces like spheres, the surface area element has to account for the curvature of the surface. For a sphere of radius \( a \), positioned at the origin, this becomes:
\