Solving an Integral Involving $\Theta$ and $u$ - Seeking Assistance

In summary, the conversation discusses how to convert a variable back to its original form in the context of integration. The solution involves making a triangle and using trigonometric functions to find the values of the variable.
  • #1
Odyssey
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Hi, I carried out the integration until the very end...I don't know how to convert the variable back to the original one. :confused:

[tex]\int_{R_{0}}^{R(\Theta)}\frac{du}{u\sqrt{a^2-u^2}} [/tex]

[tex] Let u = a\sin{\Theta}[/tex]
[tex] du = a\cos{\Theta}d\Theta[/tex]

The integral becomes...

[tex]\int_{R_{0}}^{R(\Theta)}\frac{a\cos{\Theta}d\Theta}{a\cos{\Theta}a\sin{\Theta}}[/tex]
[tex]\frac{1}{a}\int_{R_{0}}^{R(\Theta)}\csc{\Theta}d{\Theta}[/tex]

[tex]\csc{\Theta}d{\Theta} = \ln {|\csc{\Theta}-\cot{\Theta}|}+C[/tex]

This is where I'm stuck. I don't know how to convert the thetas back into the "u"s. I haven't multiplied the answer by 1/a yet. I know that [tex]\Theta=\sin^{-1}{u/a}[/tex], but if I plug the [tex]\sin^{-1}{u/a}[/tex] into Theta, the expression becomes super messy and I really don't know what to do with it.

Please help, thanks in advance! :smile:
 
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  • #2
make a triangle.. it's the only way. For example, if theta = arcsin(u/a) then sin(arcsin(u/a)) = u/a, cos(arcsin(u/a)) = sqrt(a^2-u^2)/a. Make a triangle with sides u, sqrt(a^2-u^2) and a. You can find all the trig functions from it (be sure to label theta)
 
  • #3


Hi there, it looks like you're on the right track! To convert back to the original variable, we can use the trigonometric identity \csc{\Theta} = \frac{1}{\sin{\Theta}}. Then, we can substitute in for \Theta using \Theta = \sin^{-1}{\frac{u}{a}}. This will give us:

\int_{R_{0}}^{R(\Theta)}\frac{1}{a\sin{\Theta}}d{\Theta} = \frac{1}{a}\int_{R_{0}}^{R(\Theta)}\frac{1}{\sin{\Theta}}d{\Theta} = \frac{1}{a}\int_{R_{0}}^{R(u/a)}\frac{1}{\frac{u}{a}}d{\frac{u}{a}} = \frac{1}{a}\int_{R_{0}}^{R(u/a)}\frac{1}{u}du

Then, we can use the substitution u = a\sin{\Theta} again to get back to the original variable. This will give us:

\frac{1}{a}\int_{R_{0}}^{R(u/a)}\frac{1}{u}du = \frac{1}{a}\int_{R_{0}}^{R(\Theta)}\frac{1}{a\sin{\Theta}}d{\Theta} = \frac{1}{a}\int_{R_{0}}^{R(\Theta)}\frac{1}{u\sqrt{a^2-u^2}}du

I hope this helps! Let me know if you have any other questions. Good luck with your integration!
 

1. What is an integral involving $\Theta$ and $u$?

An integral involving $\Theta$ and $u$ is a mathematical expression that involves the Greek letter $\Theta$ (theta) and the variable $u$. It is a type of integral that is commonly used in physics and engineering to solve problems involving angles and displacement.

2. How do I solve an integral involving $\Theta$ and $u$?

To solve an integral involving $\Theta$ and $u$, you need to first identify the type of integral it is (e.g. definite or indefinite), then use integration techniques such as substitution, integration by parts, or trigonometric identities to solve it.

3. What are some common mistakes when solving an integral involving $\Theta$ and $u$?

Some common mistakes when solving an integral involving $\Theta$ and $u$ include incorrect application of integration techniques, forgetting to include the variable $u$ in the final answer, and not simplifying the integral before attempting to solve it.

4. Can I use a calculator to solve an integral involving $\Theta$ and $u$?

Yes, you can use a calculator to solve an integral involving $\Theta$ and $u$. However, it is important to note that a calculator can only give you an approximate answer, and it is always recommended to check your work by hand.

5. Are there any tips for solving an integral involving $\Theta$ and $u$?

Some tips for solving an integral involving $\Theta$ and $u$ include carefully identifying the type of integral, using appropriate integration techniques, checking your work, and practicing regularly to improve your skills.

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