1.     Evaluate the definite integral from -2 to 2 of ((4-x^2)^(1/2))dx. 

Expert Answers

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Let the integral be I

`I = int_-2^2sqrt(4-x^2)dx`

this can be done using a trignometric substiution,

let `x = 2sin(u)` then `dx = 2cos(u)` and the limits,

`x =-2, sin(u) =-1 and u=(-pi)/2`

`x =2, sin(u) =1 and u=pi/2`

Therefore the integral changes,

`I = int_((-pi)/2)^(pi/2)(sqrt(4-4sin^2(u))*2*cos(u)du`

`I =4int_((-pi)/2)^(pi/2)cos^2(u)du`

but you know from trignometry, `cos(2u) = 2cos^2(u)-1`


`I =4int_((-pi)/2)^(pi/2)((1+cos(2u))/2) du`

`I =2int_((-pi)/2)^(pi/2)(1+cos(2u))du`

`2int(1+cos(2u))du = 2(u-sin(2u)/2)`


`I = 2(pi/2 - sin(pi)/2 - ((-pi)/2 - sin(-pi)/2))`

`I =2(pi/2-0+pi/2+0)`

`I =2pi`

therefore, `I = int_-2^2sqrt(4-x^2)dx = 2pi`

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