How do I evaluate this definite integral in Calc 1?

In summary, the conversation discusses evaluating the integral of a function and whether or not it is a valid integral. The integral is found to be well-defined and able to be calculated easily using symmetry. The correct form of the integral is provided and a solution is found using technology. The conversation also mentions the possibility of using trigonometric substitution to evaluate the integral.
  • #1
opticaltempest
135
0
I am trying to evaluate this integral,

[tex]\[
\int_{ - r}^r {\left( {s\sqrt {1 + \frac{{x^2 }}{{r^2 - x^2 }}} } \right)} {\rm }dx\] [/tex]

Is it a valid integral?

If I evaluate it by plugging in r and -r, it becomes undefined.
How else can I evaluate this integral? I'm in calculus 1 and I am guessing
this may be a topic covered in later calculus classes?EDIT: I had the integral wrong. It should be,

[tex]\[
\int_{ - r}^r {\left( {s\sqrt {1 + \frac{{r^2 }}{{r^2 - x^2 }}} } \right)} {\rm }dx\] [/tex]

Thanks
 
Last edited:
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  • #2
The integral is well defined (the singularity you observed is called integrable because it doesn't bother the integral), and you can calculate it really easily. Hint: think about symmetry.

Edit: Haha. Ok Halls, fairly easily. :smile:
 
Last edited:
  • #3
Well, not all that "really easily". Opticaltempest, have you covered "trigonometric substitutions" yet?
 
  • #4
I got the original integral wrong. I updated it in an edit. Sorry

I am allowed to use technology right now to evaluate the above integral.

Using Maple 10 I find the above integral evaluates to

[tex] \[
- \sqrt r \cdot \sqrt { - r} \cdot s\left( {\ln ( - r) - \ln (r)} \right)
\] [/tex]

Which simplifies to,

[tex] \[
- r \cdot s \cdot i(\pi \cdot i)
\]
[/tex]

Which simplified to,

[tex] \[
r \cdot s \cdot \pi
\]
[/tex]This was what I was looking for :) I haven't covered using trigonometric substitution yet.
I was just curious how I would integrate that without using a CAS.
 
  • #5
The form seems familiar, from what I remember it may have something to do with the arc length derivation.
 

1. What is a definite integral?

A definite integral is a mathematical concept used to find the area under a curve on a specific interval. It is denoted by ∫f(x)dx, where f(x) is the function and dx represents the infinitesimal change in the independent variable x.

2. How do I evaluate a definite integral?

To evaluate a definite integral, you can use the fundamental theorem of calculus or integration techniques such as substitution, integration by parts, or partial fractions. You can also use online calculators or software programs to find the numerical value of the integral.

3. What is the difference between a definite and indefinite integral?

A definite integral has a specific interval of integration, while an indefinite integral does not. In other words, a definite integral gives a numerical value, while an indefinite integral gives a function. The indefinite integral is also known as the antiderivative or primitive function of a given function.

4. What are some common mistakes when evaluating a definite integral?

Some common mistakes when evaluating a definite integral include incorrect limits of integration, incorrect use of integration techniques, and forgetting to add the constant of integration when finding the antiderivative. It is also important to check for any discontinuities or points of undefined values within the interval of integration.

5. When should I use a definite integral?

A definite integral is used to find the area under a curve, average value of a function, volume of a solid with known cross-sections, and solving problems involving motion and accumulation. It is also used in many real-world applications, such as calculating work, center of mass, and probability distributions.

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