Integral doble en coordenadas polares

Name: Integral doble en coordenadas polares
Uploaded: 2021-05-17T17:00:15.000Z
Duration: 18 min 1 s

Calculating the Integral of Sine in Polar Coordinates

Introduction to the Problem

The task is to calculate the integral of sin(x^2 + y^2) over region R , defined as the area in the first quadrant between two circles centered at the origin with radii 1 and 3.

Graphing the Region

A Cartesian plane is used to illustrate two circles: one with radius 1 and another with radius 3, both centered at the origin. Only the first quadrant section is considered for integration.

The relevant area is highlighted, showing it as a ring-like region between these two circles in the first quadrant.

Describing the Region in Rectangular Coordinates

In rectangular coordinates, boundaries are defined:

For x : 0 leq x leq 1

For y : bounded by curves y = sqrt1 - x^2 (upper boundary) and y = sqrt9 - x^2 (lower boundary). This makes it complex due to square roots involved.

Another part of this region is described where:

For x : 1 < x < 3

For y: similarly bounded by curves leading to further complexity in rectangular coordinates.

Transitioning to Polar Coordinates

The integral becomes complicated in rectangular coordinates; thus, switching to polar coordinates simplifies calculations.

In polar coordinates, regions resembling rings or annuli can be easily described using angles and radial distances.

Defining Angles and Radial Distances

The angle ( θ ) ranges from:

From 0 (along positive x-axis) to 90^circ or π/2. It’s crucial that angles are measured in radians for integration purposes.

The radial distance ( r ) varies between:

Inner circle radius: r = 1

Outer circle radius: r = 3. Thus, we have:

1 ≤ r ≤ 3. This effectively describes our region for integration in polar form.

Changing Variables for Integration

Conversion from Cartesian variables (x,y) to polar variables involves:

Using relationships:

x = rcos(θ),

y = rsin(θ).

Notably, since we know that

-x^2 + y^2 = r^2, this substitution simplifies our integral significantly.

Area Differential in Polar Coordinates

Unlike rectangular coordinates where differential area is simply dx dy, in polar coordinates it becomes:

Area differential: dA = r dr dθ.

This distinction is vital for correctly setting up integrals involving areas within circular regions.

Setting Up the Integral

With all transformations accounted for, we set up our integral as follows:

∫[0 to π/2] ∫[1 to 3] sin(r²) * (r dr dθ).

Here, limits reflect angular range and radial distance established earlier making computation straightforward now.

Evaluating the Integral

To evaluate this integral efficiently,

Recognize that integrating sin(r²) leads us towards using known results about trigonometric integrals.

After applying necessary substitutions and simplifications through integration techniques like u-substitution or recognizing patterns within sine functions yields manageable results.

Conclusion on Evaluation Process

As we integrate sin(r²), we find its antiderivative involves cosine functions evaluated at bounds leading us toward final answers while ensuring proper handling of constants introduced during variable changes throughout calculations.

Integration of Cosine Functions

Substituting Integration Limits

The process begins with substituting the upper limit of integration first, resulting in cos(3)^2 - cos(1)^2 . The calculations yield 9 - 1 = 8 , leading to the expression cos(9) - cos(1) .

Evaluating the Integral

The integral of a constant multiplied by the differential dtheta is simply that constant times t . This step is crucial as it confirms that we are dealing with a defined integral, which requires evaluation from an initial limit of 0 to pi/2 .

Finalizing the Calculation

When evaluating, substitute the limits into the function: first for the upper limit and then for the lower. This results in an expression involving y/2 - 0 = pi/2 , which can be multiplied by a factor of one-half present outside.

Adjusting Signs in Expressions

The negative sign can be distributed across terms, effectively flipping their signs. Thus, it transforms into -(cos(1) - cos(9)), simplifying further calculations.

Result Interpretation