Marginal PDF from joint PDF question

I am working on the 6.041 MIT Opencourseware course. Though, currently I am struggling with assignment 1b of the attached assignment (see pic).

As the joint pdf is linear (red circle), I do not see why the marginal PDF of Y can be quadratic (orange arrow). My feeling is that the provided solution mistakenly calculates the CDF.

What I did when trying to solve this; was to create a definite integral with bounds 1 and 2. In other words, replace the y (green arrow) with a 2. Though, as a result I would then have a new problem, as with this approach I would only have a constant (no variable) in my marginal PDF.

Do I have a mistake in my understanding? Thanks in advance!

• It would be beneficial if you could typeset your question plus your attempts in MathJax. General hint: The marginal pdf of a joint pdf is obtained by integrating out the other variable. Here $x$. Feb 4, 2023 at 16:23
• Please format your question using Mathjax Feb 4, 2023 at 19:49
• I think you misremembered what "marginal PDF" means. Feb 6, 2023 at 15:37

First, draw the three inequalities $$1 \leq x \leq y \leq 2$$ in the xy-plane. Then find their intersection.
To find the marginal PDF $$f_Y(y)$$, you have to integrate $$f_{X,Y}(x,y)$$ with respect to $$x$$ over the triangle depicted in the figure. Look at the arrow. It passes through $$x=1$$ first and then it passes through $$x=y$$. So $$f_Y(y)=\int_{x=1}^{x=y} \frac{3}{2}x\, dx$$ for $$1\leq y \leq 2$$.