Write the definite integral for the area of the region lying in the upper half of the ellipse given by 4x^2+y^2=4
a = ∫[-1,1] y dx
To find the definite integral for the area of the region lying in the upper half of the ellipse, we need to express the area as a definite integral using the equation of the ellipse.
The given equation is 4x^2 + y^2 = 4, which can be rewritten as y^2 = 4 - 4x^2. Now, to isolate y, we take the square root on both sides: y = ± √(4 - 4x^2).
However, we are only interested in the upper half of the ellipse, so we take the positive value of y: y = √(4 - 4x^2).
Now, to find the area, we integrate the expression for y with respect to x over the appropriate interval. In this case, the interval that covers the entire ellipse is -1 ≤ x ≤ 1, as it represents the range of x-values that satisfy the equation of the ellipse.
Therefore, the definite integral for the area of the region lying in the upper half of the ellipse is:
∫[from -1 to 1] √(4 - 4x^2) dx