Integrating (2rx - x2)1/2 w.r.t. x and Constant r

In summary, the formula for integrating (2rx - x2)1/2 w.r.t. x and Constant r is ∫(2rx - x2)1/2 dx = (2/3)(2rx - x2)3/2 + C, and the process involves using substitution and the power rule for integration. The purpose of integrating is to find the area under the curve and it has various applications in physics and mathematics. Some tips for solving these types of integrals include using substitution, trigonometric identities, and breaking the integral into smaller parts. Remember to include the constant of integration when solving.
  • #1
fysiikka111
41
0

Homework Statement


Integrate
(2rx-x2)1/2
with respect to x where r is a constant.


Homework Equations





The Attempt at a Solution


Thanks
 
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  • #2
What did you attempt already?
 
  • #3
Integration by substitution.
 
  • #4
what substitution did you try?
 
  • #5
cos(x) to see if I could then use a trig identity.
 
  • #6
Did you try to complete the square?
 
  • #7
so
(2rx-x2)1/2=(r2-(x-r)2)1/2
 
  • #8
Yes, so now it should be a lot easier...
 
  • #9
I'm not sure what substitution to use.
 
  • #11
That worked, thanks alot.
 

What is the formula for integrating (2rx - x2)1/2 w.r.t. x and Constant r?

The formula for integrating (2rx - x2)1/2 w.r.t. x and Constant r is ∫(2rx - x2)1/2 dx = (2/3)(2rx - x2)3/2 + C, where C is the constant of integration.

What is the process for integrating (2rx - x2)1/2 w.r.t. x and Constant r?

The process for integrating (2rx - x2)1/2 w.r.t. x and Constant r is to first rewrite the expression as √(2rx - x2) and then use the substitution u = 2rx - x2. This will result in du = (2r - 2x) dx. The integral can then be rewritten as ∫√u du, which can be solved using the power rule for integration.

What is the purpose of integrating (2rx - x2)1/2 w.r.t. x and Constant r?

The purpose of integrating (2rx - x2)1/2 w.r.t. x and Constant r is to find the area under the curve of the given function. This is useful for calculating physical quantities such as displacement, velocity, and acceleration in physics, as well as for solving various mathematical problems.

What are some common applications of integrating (2rx - x2)1/2 w.r.t. x and Constant r?

Some common applications of integrating (2rx - x2)1/2 w.r.t. x and Constant r include calculating the arc length of a curve, finding the volume of a solid of revolution, and determining the work done by a variable force. It is also commonly used in physics to calculate the total energy of a system.

What are some tips for solving integrals involving (2rx - x2)1/2 w.r.t. x and Constant r?

Some tips for solving integrals involving (2rx - x2)1/2 w.r.t. x and Constant r include using substitution to simplify the expression, using trigonometric identities to rewrite the integral, and breaking the integral into smaller parts if possible. It is also important to remember the constant of integration when solving these types of integrals.

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