Let R is bounded byx=y^2 and x=9. Cross section is rectangle height 2, perpendicular to x axis

Question

I am getting the wrong answer, which is 324. The correct answer is 72. I do not know if I am setting up the integral wrong.

Answer 1

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∫x^½dx. 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∫x^½dx 0≤x≤9 = 2[⅔x^3/2]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.