Calculating area of intersection I am to calculate the area of the area between the intersection of these two functions:
$x-y=7$ and $x=2y^2-y+3$
I could do this easily if the $y$ variable wasn't squared! 
I would simply set the functions on the form: $y = f(x)$ and then set $f(x) = 0$ and set the equations equal to find their point of intersection and then take the integral with the lower and upper limit of the intersections...But what do I do if I have a $y^2$? Thank you.
 A: First, determine the intersection points of the curves via substitution:
$$\begin{aligned}
(2y^2 - y + 3) - y &= 7\\
2y^2 - 2y + 3 &= 7\\
2y^2 - 2y - 4 &= 0\\
2(y^2 - y - 2) &= 0\\
2(y - 2)(y + 1) &= 0\\
y &= -1,2
\end{aligned}$$
According to the graphs of the given expressions, we construct the following integral:
$$\begin{aligned}
\text{Area} &= \int_{-1}^2 (\text{Right}(y) - \text{Left}(y))\,dy\\
&= \int_{-1}^2 \left((y + 7) - (2y^2 - y + 3)\right)\,dy
\end{aligned}$$
One constructs such integral with $y$ bounds via determining the rightmost and leftmost equations within the points of intersection.  Here, $\text{Right}(y) = y + 7$.  Solve for $x$ for $x - y = 7$ whose curve is rightmost in the $xy$-plane within the points of intersections.  Then, $\text{Left}(y) = 2y^2 - y + 3$ whose curve is the leftmost in $xy$-plane within the points of intersections.
Note: Also, as one user mentions, you can interchange $x$ and $y$ variables and then, determine the area.  Instead of having to determine the rightmost and leftmost curves within the bounds of intersections, one assigns such equations by the upper and lower curves.   Since the roles of $x$ and $y$ are interchanges, we need to integrate the difference of such equations with $x$ bounds.  For the setup I constructed, for such a case, we have
$$\text{Area} = \int_{-1}^2 (\text{Upper}(x) - \text{Lower}(x))\,dx = \int_{-1}^2 \left((x + 7) - (2x^2 - x + 3)\right)\,dx$$
which gives the same value as the integral first constructed.  Compare the graphs of these equations to the graphs of the original expressions.  The area between the intersection of the equations does not change.
Other than that, in case you want to know the answer to the problem, hover the cursor over the shaded box. :D

 The answer is $9$, which is simply found by computation.  Simplify the equation and perform the integration.  Finally, use Fundamental Theorem of Calculus.  Tada!

