Question

What change of variables is suggested by an integral containing square root 100 - x²?

A) x = 10sinθ
B) x = 10cosθ
C) x = 5sinθ
D) x = 5cosθ

Answer 1

The change of variables suggested by the integral containing the square root of (100 - x²) is x = 10cosθ.

Step-by-step explanation:

In calculus, particularly in integration, changing variables is a technique often used to simplify integrals. This technique is called substitution. The idea is to make a change of variable to transform the integral into a simpler form that is easier to evaluate.

The change of variables suggested by an integral containing the expression √(100 - x²) is x = 10cosθ. This is because we can rewrite √(100 - x²) as √(100 - (10cosθ)²) which is equivalent to √(100 - 100cos²θ). By using the identity cos²θ = (1 + cos2θ)/2, we can simplify the expression to √(100 - 100(1 + cos2θ)/2) = √(100 - 50(1 + cos2θ)) = √(50 - 50cos2θ).