Calculate the area of a region bounded by the parabola x = 9 – y 2 and the y-axis

Question

parabola  x = 9 – y 2 and the y-axis

Answer 1

This integration to find the area is best performed wrt to y. The parabola x=9-y² lies on its side with y-intercepts at y=±3. The x-axis is the line of symmetry so we only need to find the area of half the parabola, then we can double it to get the total area between the parabola and the y-axis.

Area=2∫³₀xdy=2∫³₀(9-y²)dy=2(9y-y³/3)|³₀=2(27-9)=36 square units.

This is the same as ∫³₋₃(9-y²)dy and 2∫⁹₀√(9-x)dx.