Surface integral, spherical coordinates, earth

• HmBe
In summary, the problem is to find the surface area of the Earth that lies above 50 degrees latitude North. Using the equation A = ∫R√|det(g)|dθdφ, the attempt at a solution is to integrate from 0 to 360 for φ and from 50 to 90 for θ. However, the correct limits for θ should be 0 and 40 degrees, resulting in the correct answer of 0.117 or (1-sin(50))/2. The mistake was in not taking into account that θ is equal to 90 degrees minus the latitude in the northern hemisphere.
HmBe

Homework Statement

Find the surface area of the Earth (as a fraction of the total surface of the earth) that lies above 50 degrees latitude North.

Homework Equations

$$A = \int_R\sqrt{|\det(g)|}d\theta d\phi$$

The Attempt at a Solution

Hence I get

$$\int_0^{360} \int_{50}^{90} r^2 \sin(\theta) d\theta d\phi$$

Which gives 0.321 as the answer, which just isn't right. The actual answer I know should be 0.117, or

$$\frac{1-\sin(50)}{2}$$

I'm sure something must be wrong with my integral, but I can't figure out what.

HmBe said:

Homework Statement

Find the surface area of the Earth (as a fraction of the total surface of the earth) that lies above 50 degrees latitude North.

Homework Equations

$$A = \int_R\sqrt{|\det(g)|}d\theta d\phi$$

The Attempt at a Solution

Hence I get

$$\int_0^{360} \int_{50}^{90} r^2 \sin(\theta) d\theta d\phi$$

Which gives 0.321 as the answer, which just isn't right. The actual answer I know should be 0.117, or

$$\frac{1-\sin(50)}{2}$$

I'm sure something must be wrong with my integral, but I can't figure out what.

In the northern hemisphere, the spherical polar coordinate $\theta$ is equal to $90^{\circ}$ minus the latitude. Thus the correct limits of the $\theta$ integral are 0 and $40^{\circ}$.

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You're a hero, I have mental problems. Thank you so much.

1. What is a surface integral?

A surface integral is a mathematical concept used to calculate the area of a surface in three-dimensional space. It involves integrating a function over a surface, similar to how a regular integral involves integrating a function over an interval.

2. How do you convert spherical coordinates to Cartesian coordinates?

To convert spherical coordinates (r, θ, φ) to Cartesian coordinates (x, y, z), you can use the following equations:

x = r*sin(θ)*cos(φ)

y = r*sin(θ)*sin(φ)

z = r*cos(θ)

3. How are spherical coordinates used to represent the Earth's surface?

Spherical coordinates are often used to represent locations on the Earth's surface. The radius (r) represents the distance from the center of the Earth, the inclination angle (θ) represents the latitude, and the azimuth angle (φ) represents the longitude.

4. What is the significance of the surface integral in relation to the Earth?

The surface integral is used in many applications related to the Earth, such as calculating the surface area of the Earth or finding the flux of a vector field through the Earth's surface. It is also used in geophysics to analyze data collected from the Earth's surface or to model the Earth's interior.

5. How do you calculate the surface area of the Earth using a surface integral?

To calculate the surface area of the Earth using a surface integral, you would first need to define a function that represents the Earth's surface. Then, you would integrate this function over the surface of the Earth using the appropriate limits and conversion factors. The result of the integral would be the surface area of the Earth in the desired units.

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