Integration Problem: Solve sin^3Φ from 0 to 2π

Thread starter ElDavidas
Start date Apr 23, 2006
Tags

Integration

In summary, the integration problem "Solve sin^3Φ from 0 to 2π" means finding the definite integral of the function sin^3Φ between the limits of 0 and 2π. Solving this problem is important for understanding the relationship between functions, and the steps involved include using trigonometric identities, integration techniques, and evaluation at given limits. This problem can be solved by hand or with a calculator, and it has various real-life applications in fields such as physics, engineering, and finance.

Apr 23, 2006

ElDavidas

It seems quite simple actually. But I'm still stuck:

[tex]\int_{0} ^ {2\Pi} sin^3\Phi d\Phi [/tex]

Can anyone help?

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Apr 23, 2006

HallsofIvy

Science Advisor

Homework Helper

That's a standard "sine or cosine to an odd power" integral.
[tex]sin^3\Phi d\Phi= (sin^2\Phi)(sin\Phi)d\Phi= (1- cos^2\Phi)(sin \Phi d\Phi)[/tex]
See any simple substitution you can use?

1. What does the integration problem "Solve sin^3Φ from 0 to 2π" mean?

The integration problem "Solve sin^3Φ from 0 to 2π" means to find the value of the definite integral of the function sin^3Φ, or the area under the curve of sin^3Φ, between the limits of 0 and 2π.

2. Why is it important to solve this integration problem?

Solving integration problems helps us understand the behavior and relationships between functions. In this case, solving sin^3Φ from 0 to 2π can help us understand the relationship between the sine function and its power.

3. What are the steps involved in solving this integration problem?

The steps involved in solving the integration problem "Solve sin^3Φ from 0 to 2π" include using trigonometric identities to simplify the integral, applying integration techniques such as substitution or integration by parts, and evaluating the resulting integral at the given limits.

4. Can this integration problem be solved by hand or do I need a calculator?

This integration problem can be solved by hand using basic trigonometric identities and integration techniques. However, a calculator or computer program can also be used to simplify the process.

5. What are some real-life applications of solving this integration problem?

Solving this integration problem can have real-life applications in fields such as physics, engineering, and finance. It can be used to calculate the area under a curve representing a physical phenomenon, such as the displacement of an object over time, or to determine the value of an investment over a given period of time.