Answer:
The evaluated integral in exact form is (1/2) π.
None of the option is correct.
Explanation:
To evaluate the given iterated integral by converting to polar coordinates, we need to express the integral bounds and the element of area (dA) in terms of polar coordinates.
In polar coordinates, the element of area (dA) is given by r dr dθ, where r represents the radial distance and θ represents the angle.
Let's express the integral bounds in terms of polar coordinates:
For the inner integral, the bounds of y are from √(1 - x²) to √(1 - x²), which means y stays constant.
Therefore, we can simplify the inner integral to:
∫(y stays constant) 4√(x² + y²) dy = 4y√(x² + y²)
For the outer integral, the bounds of x are from -1 to 1.
Therefore, the integral expression in polar coordinates becomes:
∫(θ=0 to 2π) ∫(r=0 to 1) (4r √(r² cos²θ + r² sin²θ)) r dr dθ
Now, let's evaluate this integral step by step:
∫(θ=0 to 2π) ∫(r=0 to 1) (4r √(r²)) r dr dθ
Simplifying:
∫(θ=0 to 2π) ∫(r=0 to 1) (4r^2) r dr dθ
Integrating with respect to r:
∫(r^2) r dr = (1/4) r^4
Now we can evaluate the inner integral:
∫(θ=0 to 2π) [(1/4) (1)^4 - (1/4) (0)^4] dθ
Simplifying:
∫(θ=0 to 2π) (1/4) dθ
Integrating with respect to θ:
(1/4) θ evaluated from 0 to 2π
Substituting the upper and lower limits:
(1/4) (2π) - (1/4) (0)
Simplifying:
(1/2) π
Therefore,
The evaluated integral in exact form is (1/2) π.
None of the option is correct.
Evaluate the iterated integral by converting to polar coordinates.
∫_(-1)^1 ∫√(1 - x²) / √(1 - x²) 4 √(x² + y²) dy dx
Enter an exact form, do not use decimal approximation.
a) 128π/3 b) 64π/3 c) 256π/3 d) 32π/3