I am having difficulty parsing this question and visualizing how the picture looks and what tools to use to solve
Question: A flat metal plate is in the shape determined by area by the area under the graph of $f(x) = \frac {1}{1+x}$ between $x = 0$ and $x = 5$. The density of the plate $x$ units from the y-axis is given by $x^2$ grams/cm$^2$. Write down a definite integral which gives the exact value of the total mass of the plate.
From what I can glean, it seems that the plate should rest horizontally on the x axis from zero to five. My first inclination was to find the center of mass, but this question is asking for me to find the total mass. What is the best approach to take with this question?
Best Answer
A small square of plate with lower left corner $(x,y)$ and upper right corner $(x+\Delta x,y+\Delta y)$ will have a mass $x^2\Delta x \Delta y$. You need to add these up over the region of interest, giving $\int_0^5\int_0^{\frac{1}{1+x}}x^2\; dy \;dx$. The $y$ integral just gives the difference of the limits because nothing depends on $y$, so you have $\int_0^5\frac{1}{1+x}x^2 \;dx$