Step by step
The question
Question: find the integral from 0 to 3 of √(9 − x²).
1 Choose the substitution
We have the root of 9 minus x².
Use x = 3 sin θ, because 1 − sin² θ is cos² θ: the minus goes away.
So x = 3 sin θ.
θ is an angle.
Its derivative: dx = 3 cos θ dθ.
Careful!
dx does not just become dθ.
dx is 3 cos θ dθ.
2 Simplify the root
Put x = 3 sin θ in: x² is 9 sin² θ.
Take out 9: its root is 3.
1 − sin² θ is cos² θ, and its root is cos θ.
3 Write it with θ
The root is 3 cos θ, and dx is 3 cos θ dθ.
The limits change: x = 0 gives θ = 0, x = 3 gives sin θ = 1, θ = π/2.
4 Integrate
The half-angle rule turns cos² θ into a plain cos.
Integrate: θ over 2, plus sin 2θ over 4.
5 Top minus bottom
At π/2: π over 4, and sin π is 0.
At 0 everything is 0.
So the integral is 9π over 4, about 7.07.
6 Check
Let's check: y = √(9 − x²) is the top of a circle of radius 3.
From 0 to 3 is a quarter of it.
Its area is π r² over 4: 9π over 4.
The same.
Right!
Let's look back.
First: choose the substitution.
Then: simplify the root.
Then: write it with θ.
Then: integrate.
Then: top minus bottom.
Last: check.