Evaluate the definite integral from 0 to 2 of √(x) / (√(x) + √(g - x)) dx.

Question image

Understand the Problem

The question is asking to evaluate a definite integral involving a square root function. The approach involves applying properties of definite integrals and performing algebraic manipulations to derive the result.

Answer

The value of the definite integral is $1$.
Answer for screen readers

The value of the definite integral is

$$ I = 1. $$

Steps to Solve

  1. Set up the equation

We want to evaluate the definite integral:

$$ I = \int_0^2 \frac{\sqrt{x}}{\sqrt{x} + \sqrt{g - x}} , dx. $$

  1. Apply the substitution

Using the substitution ( x = g - u ), the differential ( dx = -du ). The limits change from ( x = 0 ) to ( u = g ) and from ( x = 2 ) to ( u = g - 2 ):

$$ I = \int_g^{g-2} \frac{\sqrt{g - u}}{\sqrt{g - u} + \sqrt{u}} (-du) = \int_{g-2}^g \frac{\sqrt{g - u}}{\sqrt{g - u} + \sqrt{u}} , du. $$

  1. Add the two integrals

Now we have two expressions for ( I ):

$$ I = \int_0^2 \frac{\sqrt{x}}{\sqrt{x} + \sqrt{g - x}} , dx $$

and

$$ I = \int_{g-2}^g \frac{\sqrt{g - u}}{\sqrt{g - u} + \sqrt{u}} , du. $$

Adding these gives: $$ 2I = \int_0^2 \frac{\sqrt{x}}{\sqrt{x} + \sqrt{g - x}} , dx + \int_0^2 \frac{\sqrt{g - x}}{\sqrt{g - x} + \sqrt{x}} , dx. $$

  1. Combine the integrals

Adding the expressions inside the integral gives:

$$ 2I = \int_0^2 1 , dx = 2. $$

  1. Solve for ( I )

Now, divide both sides by 2:

$$ I = \frac{2}{2} = 1. $$

The value of the definite integral is

$$ I = 1. $$

More Information

This result shows how symmetry can be leveraged in integrals involving square roots. The function's behavior around the midpoint of the interval often allows simplification.

Tips

  • Forgetting to change the limits when performing substitutions can lead to incorrect evaluations.
  • Doing algebraic manipulations incorrectly, especially when combining integrals.

AI-generated content may contain errors. Please verify critical information

Thank you for voting!
Use Quizgecko on...
Browser
Browser