Main Answer:
The volume of the solid under the hemisphere z = (1 - x² - y²)¹/² and above the region bounded by the graph of the circle x² + y² - y = 0 is (1/2)π.
Step-by-step explanation:
To find the volume, we integrate the given hemisphere equation over the region defined by the circle equation. Firstly, convert the circle equation to polar coordinates by expressing x and y in terms of r and θ. The bounds for the integral in polar coordinates are determined by the circle's intersection points with the x-axis. Then, set up the double integral with the polar coordinate limits, taking the square root of the hemisphere equation as the upper limit.
Simplify and solve the double integral, incorporating the polar coordinate transformation. The resulting expression is (1/2)π, representing the volume of the described solid. The integration process involves carefully handling polar coordinate substitutions and manipulating the equations to fit the coordinate system. This simplification leads to the concise answer of (1/2)π for the volume of the specified solid.