How Do You Calculate the Area Bounded by \( r = 8\cos(10\Theta) \)?

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Homework Help Overview

The problem involves calculating the area of the region bounded by the polar equation \( r = 8\cos(10\Theta) \), which describes a rose curve with multiple petals.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss setting \( r = 0 \) to determine bounds for \( \Theta \) and the use of integration to find the area. There is a mention of multiplying the area found for one petal by 20 to account for all petals. Questions arise regarding the evaluation of the integral and the correctness of the steps taken in the calculations.

Discussion Status

The discussion includes attempts to clarify the integration process and the application of trigonometric identities. Some participants express uncertainty about the evaluation of the integral and the correctness of their bounds, while others provide guidance on these aspects. There is no explicit consensus on the final solution, but productive dialogue is ongoing.

Contextual Notes

Participants are working under the constraints of a homework assignment, which may limit the information they can share or the methods they can use. There is a focus on ensuring the correct interpretation of the problem setup and the mathematical principles involved.

Cici2006
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Homework Statement


Find the area of the region bounded by r=8cos10[tex]\Theta[/tex]


Homework Equations





The Attempt at a Solution



I set r=0 to find [tex]\Theta[/tex], which i used for my bounds
[tex]\Theta[/tex]=pi/20, 3pi/20
A= [tex]\int[/tex](1/2)64cos^2(10[tex]\Theta[/tex]) d[tex]\Theta[/tex]
 
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What you need to do is multiply your answer by 20 since you found the area of one of the 20 petals of the rose curve.
 
Are you having trouble finding the correct solution since yours is too small? or because you don't know how to evaluate the integral?

If you need help evaluating the integral, use the fact that

[tex]\cos^2{nx} = \frac{1+\cos{2nx}}{2}, n\in\mathbb{N}[/tex]
 
Okay, let me state what i did in more detail:
A=(1/2)integral 64(cos^2(10theta)) d(theta)
=32 integral (1/2)(1+cos20theta) (theta)
=16[theta-(1/20)sin20theta]
did i do it correct so far?
then i just plug in my bounds which are pi/20 to 3pi/20 right?
now should i just multiply my answer by 20?
 
thanks for the help i solved it
 

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