I know that this must maximize the definite integral from 0 to the x value, which would use the derivative of the integral but I'm unsure of how to set up the equation.
2026-05-17 08:30:17.1779006617
Find the dimensions of the rectangle with max area, base on positive x-axis, a side on the y-axis, and a vertex on y = e^(−x^2)
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If the base is $x$, the height is $e^{-x^2}$.
Do you see why?
Then, what is the area?
Can you find the extreme value of the area.
Can you do this for an arbitrary function $f(x)$ instead of $e^{-x^2}$?
Are there some $f(x)$ for which there is no maximum area?
Are there some $f(x)$ for which there are multiple maximum areas?
What happens when $f(x) = e^{-x}$?
What happens when $f(x) = \sin(x)$?