Question
Given a function f(x)=sqrt(100-x^2)
Evaluate the integral xsqrt(100-x^2dx) for the interval (0,10)
Evaluate the integral xsqrt(100-x^2dx) for the interval (0,10)
Answers
Steve
let x = 10sinθ. then
dx = 10cosθ dθ
√(100-x^2) = √(100-100sin^2θ = 10√(1-sin^2θ) = 10cosθ
The integral then becomes
∫(10cosθ)(10cosθ dθ) = 100∫cos^2 θ dθ
which I'm sure you can do.
The limits of integration then become
0<=x<=10 --> 0 <= θ <= π/2
dx = 10cosθ dθ
√(100-x^2) = √(100-100sin^2θ = 10√(1-sin^2θ) = 10cosθ
The integral then becomes
∫(10cosθ)(10cosθ dθ) = 100∫cos^2 θ dθ
which I'm sure you can do.
The limits of integration then become
0<=x<=10 --> 0 <= θ <= π/2