R is bounded by x=y2 and the line x=9. x=y2 is a sideways parabola passing through the origin. Half the parabola lies below the x-axis. The line meets the parabola when 9=y2, that is, at (9,3) and (9,-3). If we use the integral ∫ydx for 0≤x≤9 we have to use ∫√xdx which gives us only half the area because y=√x is only the positive half of a parabola. So we need to double this to get the whole area=2∫x½dx. The function x=y2 has no inverse by the definition of function: neither the function y=√x or y=-√x is the inverse function because we get only half a parabola.
2∫x½dx 0≤x≤9 = 2[⅔x3/2]09=(4/3)27=36 square units.
Now we need to consider the volume of the solid prism which has a parabolic base with the area calculated above and a height of 2 units. So the volume is 2×36=72 cubic units.
There is an alternative way of calculating this. If we reverse the roles of y and x (90° rotation and reflection), we get y=x2 and the line y=9. The rectangular cross-section is then perpendicular to the y-axis. However if we use the integral ∫x2dx for -3≤x≤3 we get the result 18, because this is the area between the parabola and the x-axis, whereas we require the area between the parabola and the y-axis. To get the correct area we need to subtract it from the area of the enclosing rectangle with dimensions 9×6=54, so 54-18=36, where 9 is the height of the parabola from the origin and 6 is its width (3-(-3)). Then the volume of the prism becomes 72 cubic units.