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Studying for a test? Prepare with these 8 lessons on Applications of definite integrals.

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# Worked example: problem involving definite integral (graphical)

Video transcript

Voiceover: It took 20
minutes before Jughead noticed that his hot
tub had sprung a leak. Once he realized it, he
opened the hot tub's drain and the rest of the water
rushed out in 40 minutes. The rate at which the water
drained from the hot tub in gallons per minute is shown. How many gallons of
water were in the hot tub before it started to leak? So let's see what they have over here. So they've plotted gallons per minute versus time in minutes. So we see here, this blue line, this is the rate at which
water drained from the hot tub. So at minute zero, the
water wasn't really draining from the hot tub and then
not just more drained but the rate at which the
water was draining increased. So 10 minutes into his
bath, the water was draining at a gallon per minute and
then 20 minutes into his bath water was draining at
two gallons per minute. Then he noticed it and he opens the drain. I guess he wants to accelerate
the end of his bath. So he opens the drain
and then all of a sudden water starts draining out
at a much higher rate, at 20 gallons per minute,
but then that decreases. So we could think about physically
why that might decrease. Maybe there was just less
pressure or whatever. We're not going to go
into the physics of it, but we're just going to
take this chart as fact. The rate at which the
water drains decreases all the way to the 60th minute, which is 40 minutes after
he opened the drain. At the 60th minute, all of the water was actually drained out. So given that, how do we think about how many gallons of
water were in the hot tub before it started to leak? And I encourage you to pause the video and try to see if you could
figure it out yourself before I work through it. Well, to answer this question: How many gallons of
water were in the hot tub before it started to leak? That is answering the same question as How much total hot water drained out? So how much drained out? And to think about that,
we can just kind of go back to what we knew before we
learned about calculus. If I have something
happening at a fixed rate, so let's say that this
is gallons per minute. So this is still the same
context, I guess you could say. This is time in minutes. Let's say things are draining
out at a constant rate. If you wanted to figure
out how much drains out over a certain interval of time, let's say this interval
of time right over here, let's call that delta t, you would just multiply the
rate over that interval of time, which we could represent by this orange height right over here, times the amount of time that passed by, which would give you the area under the curve over that interval. the area under the curve
over that interval. This area would tell you the gallons that drained over that delta t. And this doesn't just apply
when you have a constant rate. If your rate looked something like this, as we've seen in other videos, you could figure out the
amount that has drained in a specific interval, let's say this interval right over here, by essentially figuring out
the area under the curve over that interval,
and the units work out. If you multiply gallons
per minute times minute, this area is going to
be in terms of gallons. Another way you could think about it, and this goes back to,
well, how do you figure out the area of a trapezoid right over here. Well, the area of a trapezoid, you could find the average
height of the trapezoid, which would be the average of the beginning and the end period, you take that average over there, and this would work for a line like this, if you take that average height and multiply by the change in time, you are going to figure out that area. And that is another way
of thinking about it. You are taking the average
rate over that interval times the interval, and
that is going to give you the total number of gallons. And so we just have to
apply that idea over here. We just have to literally figure out the area under the curve
over the entire interval when the water was actually
leaking or draining. So essentially the area under the curve between zero minutes and 60 minutes. And so it is going to be this area plus all of this area under this part as well. To help us think about that, I'm going to just split
that up into some sections. So I'll have this
triangular section up here. I could just think about
this as a trapezoid, but I'm just going to split it up into a triangle and a rectangle. And then I have this section
right over here in green. So what is the area of this entire thing? Well, the area right over here, we have 20 minutes times two
gallons per minute times 1/2. This gives us the area
under this triangle. So that's going to be 20 gallons. We see that the units work out nicely. So that is essentially
how much has drained out in the first 20 minutes. And then this green area
is going to be 40 minutes times 10 gallons per minute. And actually maybe you know the units, since I'm breaking it
up in this strange way, I'll just figure out this
area in a unit-less way. So 40 times 10, which is equal to 400. And then, finally, in blue,
I have 40 times this height right over here between
10 and 20 is another 10 but then I'm going to
multiply that times 1/2. So it's going to be 40 times 10 times 1/2, which is going to be 200. And so when you add all
of these areas together, you are going to get 400
plus 200 is 600 plus 20. You are going to have 620 gallons. 620 gallons is how much
water in total drained out or how much water was in the
tub before it started to leak, or the hot tub, yeah this is
quite big for a regular tub, bu this could be like a Jacuzzi
or a hot tub or something.