Integrating Problem: Definite Integral from sqrt2 to 2

In summary, the person is asking for help with a definite integral involving the function 1/x^3*sqrt(x^2-1). They have separated the integral into two parts and are attempting to use a trigonometric substitution for the second part before integrating it. They are unsure if they are going about it correctly and are seeking clarification.
  • #1
miller8605
17
0

Homework Statement


i'm taking a defitinite integral from sqrt2 to 2 of the function 1/x^3*sqrt(x^2-1)dx.

Homework Equations





The Attempt at a Solution


I separated it into 1/x^3 and 1/sqrt(x^2-1). I have the second part using trig sub. as being sec theta dtheta, before integrating it. I believe i did this part correctly.

What I can't remember is that I make 1/x^3 to x^-3 and then integrate it that way with the final being -1/2(1/x^2)??

Am i going about this correctly?
 
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  • #2
miller8605 said:
I separated it into 1/x^3 and 1/sqrt(x^2-1).

No need to do that. The entire integrand comes out very nice with a trig substitution.

I have the second part using trig sub. as being sec theta dtheta, before integrating it. I believe i did this part correctly.

The part in red makes very little sense to me, but it sounds like it has the kernel of a correct method in there. Could you elaborate?
 

1. What is the purpose of integrating a definite integral from sqrt2 to 2?

The purpose of integrating a definite integral from sqrt2 to 2 is to find the exact area between the curve and the x-axis within that interval. This can be useful in solving real-world problems involving rates, distances, and other quantities represented by a function.

2. How is the definite integral from sqrt2 to 2 calculated?

The definite integral from sqrt2 to 2 is calculated using the fundamental theorem of calculus, which states that the definite integral of a function can be calculated by finding its antiderivative and evaluating it at the limits of integration. In this case, the antiderivative of the function within the given interval is used to find the exact area.

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

A definite integral has specific limits of integration, while an indefinite integral does not. A definite integral gives a numerical value, while an indefinite integral yields a function. Essentially, a definite integral represents the exact area under a curve, while an indefinite integral represents the general equation of the curve.

4. Can the definite integral from sqrt2 to 2 have a negative value?

Yes, the definite integral from sqrt2 to 2 can have a negative value. This occurs when the area under the curve is below the x-axis, resulting in a negative value for the definite integral. It is important to consider the direction of the function and the limits of integration when interpreting the sign of the definite integral.

5. What are some real-world applications of integrating a definite integral from sqrt2 to 2?

Integrating a definite integral from sqrt2 to 2 can be used in various real-world applications, such as calculating the work done by a variable force, finding the displacement of an object with changing velocity, and determining the volume of irregularly shaped objects. It can also be used in economics, physics, and other fields to analyze rates of change or to solve optimization problems.

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