Let's say that I can fill one bucket of water each day

1 dt
t=# days

How many buckets of water will I have at any given time?

integral(1 dt)

Think about the relationship between the time and number of buckets. What's the relationship? 1 to 1 also known as linear

t

What else?
A little tricky but I also didn't say how many buckets of water I started with, so that's another variable c

integral(1 dt) = t + c

You've just performed an integral

So how many buckets of water will I fill between now and 10 days from now: Definite integral

integral 0 -> 10 (1 dt) = (10 + c) - (0 + c) = 10 + c - c = 10

Let's say I have 14, 13, 12, 11, 10 buckets of water as each day goes by

y(t) = 14 - t

How quickly is the number of buckets of water changing over time?

d/dt(y) = -1

I'm losing one bucket each day