The area of the region bounded by the graphs of the equations is a√π.
Sure, here is the solution to find the area of the region bounded by the graphs of the equations using an iterated integral
The given equation represents an ellipse centered at the origin with semi-major axis a and semi-minor axis b. The area of an ellipse is given by the formula:
Area = πab
To find the area using an iterated integral, we can convert the ellipse into bounds for a double integral. Since the ellipse is symmetric about the x and y axes, we can integrate over half of the ellipse and multiply by 2.
We can express the ellipse in terms of y by solving for x:
x = ±√(a² - y²/b²)
Now we can set up the double integral:
Area = 2 ∫[-b, b] √(a² - y²/b²) dy
Evaluating the integral:
Area = 2 ∫[-b, b] √(a² - y²/b²) dy
= 2a√(π/2)
= a√π