I'm having trouble getting this through my head. I have an exercise in which I'm told that $f(x) = \sqrt x $ and $ g(x) = 2-x $. Using definite integrals, I'm supposed to find the area of the region under the two functions. This is fairly simple, it's not even the problem.
I've included an image of the graph and a link to the Desmos graph at the bottom of the page.
Before telling me to discover the area, they ask me to formulate it with integrals using the $x$ variable which I did, stating that the region could be represented by $ \int_0^1 f(x)dx + \int_1^2 g(x)dx$, and this answer is completely correct. The problem is that they then tell me to do the same using the $y$ variable, and the supposed correct answer is $\int_0^1 (2-y)dy - \int_0^1y^2dy$, and I cannot even visualize this in my mind. How can this be true? It checks out numerically, but how can this be correct?
*Edit: Also, since I'm having trouble visualizing the region in $y$-representation, is there a simple way to obtain it from the $x$-representation?
Resources:
$\int_0^1 (2-y)dy$ is the area of the red, blue, and purple areas;
to get the area you want, subtract the red area, which is $\int_0^1 y^2 dy$.