i know how the function looks like when graphed because i used a graphing calculator... but what does y=0 mean???
Hey asilvester635.
Setting y = 0 means that you are finding the solutions of when the function crosses the x-axis.
This is also known as finding the roots of the equation where f(x) = -x^2 + 3x = 0.
Finding roots to equations has many purposes.
Start by factorizing the equation into (x-a)(x-b) = 0 and then use the fact that x = a and x = b are solutions to this equation since at least one term has to be 0 in order for the LHS to be zero.
$\displaystyle y = 0$ is the x-axis
so, you're looking for the area between the curve $\displaystyle y = 3x-x^2$ and the x-axis
note the curve intersects the x-axis when $\displaystyle 3x-x^2 = 0$
$\displaystyle x(3-x) = 0$
... the zeros, $\displaystyle x = 0$ and $\displaystyle x = 3$ are the limits of integration.