Solving a Polar System Integral - Can You Help?

In summary, a polar system integral is used to solve problems in polar coordinate systems by calculating the area under a curve. To solve it, one can identify the function and limits of integration, convert the function to rectangular coordinates, integrate using standard techniques, and substitute the limits back into the solution. Common techniques include using trigonometric identities and symmetry, and it has applications in physics, engineering, and other scientific fields. An example of solving a polar system integral is shown with the function ∫<span style="vertical-align:super; font-size:smaller;">0</span><span style="vertical-align:sub; font-size:smaller;">π/2</span>∫<sub>0</sub><sup>
  • #1
b0t2
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I have a task to solve it in polar system: x²=4y-y²; x²=8y-y²; y=x; x=0. So in polar: r=4sin phi; r=8sin phi; phi=pi/4; phi=pi/2. The integral - int(from pi/4 to pi/2) d phi int(from 4 sin phi to 8 sin phi) r dr. My answer is 3pi - 1/4 but seems like its not true. Somebody has another answer?
 
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  • #2
b0t2 said:
I have a task to solve it in polar system: x²=4y-y²; x²=8y-y²; y=x; x=0. So in polar: r=4sin phi; r=8sin phi; phi=pi/4; phi=pi/2. The integral - int(from pi/4 to pi/2) d phi int(from 4 sin phi to 8 sin phi) r dr. My answer is 3pi - 1/4 but seems like its not true. Somebody has another answer?

Welcome to MHB, b0t2! :)

Let's rewrite your statements in $\LaTeX$.
Apparently your problem has:
$$x^2=4y-y^2; \quad x^2=8y-y^2; \quad y=x; \quad x=0$$
So in polar:
$$r=4\sin \phi; \quad r=8\sin \phi; \quad \phi=\pi/4; \quad \phi=\pi/2$$
The integral:
$$\int_{\pi/4}^{\pi/2} d \phi \int_{4 \sin \phi}^{8 \sin \phi} r dr$$
Your answer is $3\pi - 1/4$ which appears not to be true.

Perhaps you can clarify?
From your first statement I deduce x=y=0, but obviously that is not what you intended.
I suspect you're talking about some kind of intersection of surfaces, but I prefer not to guess.
Do you perhaps have the complete problem statement?
 

FAQ: Solving a Polar System Integral - Can You Help?

What is a polar system integral?

A polar system integral is a type of integral used to solve problems in polar coordinate systems. It involves calculating the area under a curve in a polar coordinate system, which is different from calculating the area under a curve in a Cartesian coordinate system.

How do you solve a polar system integral?

To solve a polar system integral, you can follow these steps:

  • 1. Identify the function and the limits of integration. The function will typically be in terms of r and θ.
  • 2. Convert the function from polar coordinates to rectangular coordinates using the conversion formulas.
  • 3. Integrate the function using standard integration techniques.
  • 4. Substitute the limits of integration back into the solution.

What are some common techniques used to solve polar system integrals?

Some common techniques used to solve polar system integrals include:

  • Substituting r and θ with their corresponding rectangular coordinates using the conversion formulas.
  • Using trigonometric identities to simplify the integrand.
  • Using symmetry to simplify the integral.
  • Breaking the integral into smaller parts and applying different techniques to each part.

What are some applications of polar system integrals?

Polar system integrals have various applications in physics, engineering, and other scientific fields. Some examples include calculating the moment of inertia of an object, finding the center of mass of a system, and calculating the electric field due to a charged disk.

Can you provide an example of solving a polar system integral?

Yes, here is an example of solving the polar system integral ∫0π/202 r³ cos(θ) dr dθ:

  • Step 1: Convert the function to rectangular coordinates: ∫0π/202 (r³ cos(θ)) dr dθ = ∫0π/202 (x³ cos(θ)) dx dθ
  • Step 2: Integrate the function: ∫0π/202 (x³ cos(θ)) dx dθ = ∫0π/2 [x⁴/4] 02 cos(θ) dθ = ∫0π/2 4cos(θ) dθ = 4sin(θ) 0π/2 = 4

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