Calculus 5.2 Definite Integrals

Question

For this problem I was given a right Riemann Sum. The sum being

∫[√(4-(2/n)²)*(2/n)] + [√(4-(4/n)²)*(2/n)] + ... + [√(4-(2n/n)²)*(2/n)]

The definite integral given is ∫ with a=0 and b=?

a) I'm asked to find the value of b

b) What f(x) is equal to

c) The limit of the Riemann sums as n -> ∞.

Based on the change of x being 2/n I correctly found put b=2.

For f(x) I noticed each subpoint has an f(x) of √(4-x²), also giving a correct answer.

The last question is where I'm more confused. Based on f(x) we have a semi circle. I noticed it maxes out at 2 and that was wronged. I checked videos and my textbook for examples. They were very long and confusing for me. I was hoping I can get a simple explanation on how to solve the last part step by step. Thank you and have a good day.

Answer 1

The graph of x²+y² = 4 is the circle with center (0,0) and radius2. Solving for y, we get y = ±√(4 - x²).

∫_{(from 0 to 2)} √(4 - x²)dx = Area of the upper right quarter of the circle = (1/4)π(2)² = π.