So I would like to ask how exactly do we determine what limits to take when integrating both Cartesian and parametric equations.
So let's say we have a graph of $y=x^2$. If we wanted to take the area between $0$ and $5$, would we take limits as $5$ on the top and $0$ on the bottom or the other way round. What does the top number signify and what does the bottom number signify.
What about the area between $-5$ and $-3$. What would we put on the top and what would we put on the bottom, Why?
Let's say we were to find Volume of a revolution of the parametric equations
$$x=(t^2)^{-1}$$ $$y=(e^t)$$
from the ordinate of $P$ where $t=0$ to the ordinate of $Q$ where $t=-1$. Would we take $0$ on top and $-1$ on the bottom, If not, why??? What do the limits signify in terms of area?
Could somebody please explain this concept? I tried googling this topic but none explained what to take in terms of limits of an intergral and why we take limits in the way we do. I heard its something from left to right? Not very sure if that is correct and applies to all quadrants of the graph and not sure why we do that.
Thanks
A definite integral has two limits, an upper limit and a lower limit, hence a "top number" and a "bottom number". In many cases, the upper bound of the interval over which an integral is taken is assigned to be the upper limit of the integral, and the lower bound becomes the lower limit of the integral. In fact, one may always follow that procedure as a rule, for there is a simple relation between a definite integral and an integral in which only the limits have been interchanged: the definite integral from a to b is the negative of the same integral from b to a. Thus, the area of the plane bounded by the curves f(x) = x^2, x = 0, x = 5, and y = 0, can be measured by integrating from 0 to 5 or by integrating from 5 to 0, and then taking the negative of the result.