Thread: Related Rates and finding are enclosed by curve help

The Questions:
1). Find Area of the region enclosed by the y-axis and the curves y=x^2 and y=(x^2+x+1)e^-x

2). Coffee is draining from a conical filter into a cylindrical coffeepot at the rate of 10in^3/min.
Then there's a picture of a 6 in. diameter cone, and it also has a height of 6 in. Then the pointy end of the cone is facing down touching a cylinder with a 6 in. diameter. (no height)
How fast is the level in the pot rising when the coffee in the cone is 5 in. deep?
How fast is the level in the cone falling at that moment?

So for the first one, the only I idea i have it to find the point when the 2 equations intersects, would be nice if I can do that without a calculator if possible, then I integrate both equations. Then plug in x for what value it was for when it the equations intersected. Subtract the values of the integral of y=(x^2+x+1)e^-x and y=2x to get my answer, I hope. Please clarify to me if this works.

The second one I'm more clueless in (haven't done this in a while). Ok, so I'm trying to find dh/dt, and I know my dV/dt is -10in.^3/min. Afterward I'm lost in what to do after that. Please explain work and show me how this is done, thank you.

1) Where are you struggling? Finding the intersection of the two functions? Evaluating the integral? Somewhere else?