Final answer:
To find the area of the region inside the polar curve r = 7 + 4 sin θ below the polar axis, one must set up a polar integral with appropriate bounds for θ, corresponding to the curve's placement below the x-axis, and integrate ½ r² dθ over those bounds.
Step-by-step explanation:
The question asks for the area of the region inside the polar curve r = 7 + 4 sin θ that lies below the polar axis. When finding the area below the polar axis, one technique is to utilize symmetric properties if applicable and adjust the limits of integration accordingly. If the curve is symmetric about the horizontal axis (polar axis), then the area below can often be calculated as half of the entire area of the curve.
However, this specific function does not display symmetry that simplifies the problem in such a way. The area below the polar axis for this curve would require setting up a polar integral with appropriate bounds for θ, taking values that correspond with the curve existing below the polar axis. This usually involves angles from π to 2π, as these are the angles where the sine function is negative (or zero), reflecting that the curve dips below the x-axis.
However, without performing the actual integration, a precise answer cannot be provided here. For a full solution, one would need to integrate ½ r² dθ over the correct interval for θ, which represents the infinitesimally small area elements of the curve below the polar axis.