The setup for these problems is really the key to the whole thing. Once you have your integral setup, you only need to run the standard Definite Integral process.
Step 1: Determine whether your equation is in terms of x (standard: $y={\textcolor[rgb]{}{x}}^{2}+3\textcolor[rgb]{}{x}+\sqrt{\textcolor[rgb]{}{x}}$) or in terms of y (nonstandard: $x={\textcolor[rgb]{}{y}}^{2}+3\textcolor[rgb]{}{y}+\sqrt{\textcolor[rgb]{}{y}}$).
This is one of the few times in AP Calculus that you have to consider the graphical changes that occur when you change the independent variable . The reason it is important is that your Area Between Two Curves setup requires you to know which of your equations is on top and which one is on bottom, and how you visualize those will change depending upon which variable your equation is “ in terms of ”, x or y .
Standard setup with equations in terms of x , dx . The majority of equations you have seen to this point will have been in the form of a y = x ’s (i.e., $\textcolor[rgb]{}{y}=\sqrt{\textcolor[rgb]{}{x}}$). I know that you have used different variables, especially with “realworld” problems, but they have always been of the format where the xaxis is the independent variable (what you plug in ), and the yaxis is the dependent variable (what you get out ).

New setup with equations in terms of y , dy . For the first time in the AP Calc course, they start introducing situations where the yvariable is the independent variable (what you plug in ), and the xvariable is the dependent variable (what you get out ). You have equations now where x = y ’s (i.e., $\textcolor[rgb]{}{x}={\textcolor[rgb]{}{y}}^{2}+2\textcolor[rgb]{}{y}\u20137$).
As you try to determine which equation is on top on bottom, I have found it useful to physically rotate my book, paper, or calculator 90 ^{◦} to the left (counter clockwise) when first learning the process. Some people are great at doing these types of rotations in their heads, and you will be able to too after some practice. When you first start though you might want to try it. 

Standard: Not Rotated
xaxis running left and right yaxis running up and down 
Rotated left 90 ^{◦}
xaxis is running up and down yaxis is running left and right


$\underset{\textcolor[rgb]{}{a}}{\overset{\textcolor[rgb]{}{b}}{\int}}\textcolor[rgb]{}{\mathit{Top}}\textcolor[rgb]{}{\u2013}\left(\textcolor[rgb]{}{\mathit{Bottom}}\right)\textcolor[rgb]{}{}\textcolor[rgb]{}{\mathit{dy}}$ 
Step 2: Determine the bounds of the integral, $\underset{\textcolor[rgb]{}{a}}{\overset{\textcolor[rgb]{}{b}}{\int}}$, for each of your enclosed regions.
Intersections as Bounds

Equations as Bounds

Step 3: Determine the Top and Bottom equations for all of your enclosed regions.
The short list of graphs you should be able to sketch a graph on your own would be:
That is not an exhaustive list, but will give you a good starting point.
Create a number line broken into regions based upon the intersections or bounds that you found in Step 2 . (Think number line like a 1 ^{st} Derivative Test). Choose a test xvalue from each of the regions you have laid out.
The larger yvalue represents the graph that is on Top because its yvalue is larger than ( above ) the other equation’s yvalue .
Step 4: Setup a Definite Integral problem for each of your enclosed regions using your Area Between Two Curves formula :
$\underset{\textcolor[rgb]{}{a}}{\overset{\textcolor[rgb]{}{b}}{\int}}\textcolor[rgb]{}{\mathit{Top}}\u2013\left(\textcolor[rgb]{}{\mathit{Bottom}}\right)\mathit{dx}=\textcolor[rgb]{}{\mathit{Area\; Betwen}}\textcolor[rgb]{}{2}\textcolor[rgb]{}{\mathit{Curves}}$
Remember if you have multiple regions, then you will need to add up all those results to get your final answer for the Total Area Enclosed.