So I have the following problem:
A vessel is in the form of an inverted circular cone with a semi-vertical angle of 30 degrees. Water is poured in at 5cm^3/min and leaks out from a hole at the bottom at a rate of cm^3/min. Set up the integral that gives the time for the depth to change from 0.5cm to 5cm.
I have got the (correct) integral:
Then the answers say I should take the integral of the above equation, with respect to h, from h=0.5 to h=5.
I'm wondering, why? Doesn't taking that integral just find the area of the curve? How does it relate to the question? Or does doing that subtract the time at t=5 from the time at t=0.5?
Just confused about the why. Thanks!