Answer:
V = π (rh² − ⅓h³)
Step-by-step explanation:
Draw a cross section of the bowl. Cut a thin, horizontal slice of the water. This slice is a circular disc of radius x and thickness dy. It is position a distance of y from the bottom of the bowl. The volume of this slice is:
dV = πx² dy
By drawing a right triangle, we can define x in terms of y:
x² + (r−y)² = r²
x² + r² − 2ry + y² = r²
x² = 2ry − y²
Substitute:
dV = π (2ry − y²) dy
The total volume of the water is the sum of all the slices from y=0 to y=h.
V = ∫₀ʰ π (2ry − y²) dy
V = π ∫₀ʰ (2ry − y²) dy
V = π (ry² − ⅓y³) |₀ʰ
V = π (rh² − ⅓h³)
