Find the integral?

Question

Find the integral of sqrt(4-x^2)dx from 0 to 2. 

 

Answer: pi

 

Please show all your work.

Answer 1

As a check on a calculus solution, y = √(4 - x^2) describes a semicircle in above the x-axis centered at the origin with radius 2.

 

The integral of sqrt(4-x^2)dx from 0 to 2 will be the area of 1/4 the full circle, which is pi 2^2/4 = pi.

Answer 2

 2                1/2

∫  (4 - X^2 )  dx

0

 

   X/2 = Sin U    X = 2 Sin U   dx = -2 CosU

 

   ( 4 - X^2 )^1/2  = Cos U

/ Graph a triangle with hypotenuse 2, and opposite side to angle u is X.

 

2

∫  ( 4 - X^2 ) ^ 1/2 dx = ∫- 2 Cos ²u du +c =

0

      

      2Cos²U  = Cos 2U +1

  

      Sin^-11 

- 2 ∫          Cos 2 U +  2U  =

    Sin^-1(0)

 

   4 Sin2U₀^π/2 + 2∏/2 - 0

 

     0  + ∏ = ∏