Consider a 3d rectangular coordinate system.
A semicircle of radius a is drawn on the x-axis with its centre at the origin. An identical circle is drawn upon the y-axis. The two circles are oriented perpendicular to one another.
If we use Riemann's sums, multiplying the two curves with one another should give us a 3D hemisphere on the x-y plane.
The areas under each 2D curve is [MATH]\pi\frac{a^2}{2}[/MATH]The volume of the hemisphere should be the product of the areas under the two curves, which it obviously isn't.
Where am I going wrong?
A semicircle of radius a is drawn on the x-axis with its centre at the origin. An identical circle is drawn upon the y-axis. The two circles are oriented perpendicular to one another.
If we use Riemann's sums, multiplying the two curves with one another should give us a 3D hemisphere on the x-y plane.
The areas under each 2D curve is [MATH]\pi\frac{a^2}{2}[/MATH]The volume of the hemisphere should be the product of the areas under the two curves, which it obviously isn't.
Where am I going wrong?