Consider a region between the 2 following equations... $x= (y-3)^2 + 3$ and $y= -x^2+5$ bounded by the horizontal lines $y=5$ and $y= -1$. Set up the integral using the y-axis and then set up an integral using the x-axis. Finally, compute and compare the values of both integrals
Here is my potential approach to getting the y-axis integral...
$$\int_{-1}^{5}[(y-3)^2+3-\sqrt{5-y}]dy$$
Im not sure how to get the x integral
First find intersection points: $$-{ x }^{ 2 }+5=\sqrt { x } +3\\ x_{ 1 }=1\\ -{ x }^{ 2 }+5=-\sqrt { x } +3\\ { x }_{ 2 }=1.831$$ And your integral will be $$\\ \int _{ 0 }^{ 1 }{ \left( \sqrt { x } +3 \right) -\left( -\sqrt { x } +3 \right) } dx+\int _{ 1 }^{ 1.831 }{ \left( -x^{ 2 }+5 \right) -\left( -\sqrt { x } +3 \right) dx } $$