Final answer:
The area of the plane figure bounded by the curve x=2cost y=6sint and the Ox axis (y≥0) is equal to k⋅π. To find k, we need to find the area of the region under the curve. This can be done by integrating the curve with respect to x from the point where it intersects the Ox axis to where it intersects again.
Step-by-step explanation:
The area of the plane figure bounded by the curve x=2cost and y=6sint and the Ox axis (y≥0) is equal to k⋅π. To find k, we need to find the area of the region under the curve. This can be done by integrating the curve with respect to x from the point where it intersects the Ox axis to where it intersects again.
The equation can be rewritten as y = 6sin(t) and x = 2cos(t). We need to find the values of t where y is greater than or equal to zero, that is, when sin(t) is greater than or equal to zero. This occurs in the interval t ∈ [0, π].
Taking the integral of y with respect to x in this interval gives us the area:
- ∫0π 6sin(t) dt
- = -6cos(t) ∣0π
- = -6cos(π) - (-6cos(0))
- = 12
Therefore, k is equal to 12, and the area of the plane figure bounded by the curve and the Ox axis is 12π.