Hi, everyone. Welcome back to integralcalc.com. Today we’re going to be doing a double integrals
problem and the one we’re going to be working on is the integral from -1 to 2 of the integral
from 1 to 3 of 2x – 7y dy dx. So the crucial thing here is the order of
dy dx. What it tells us is that the first thing we want to pay attention to is the inner
integral. We can draw brackets around it like this but we’ve got an inner integral and
an outer integral. And we want to integrate from the inside out. So we’re going to be
paying attention to everything that’s inside the inner integral. It tells us that we’re
going to take the integral first with respect to y because we see dy as the inner terminology.
dx is on the outside. So because dy is in the inside, we’re taking the integral with
respect to y first of 2x – 7y from 1 to 3. Once we take the integral, evaluate on
the range 1 to 3 and simplify, we’ll put that inside of the outer integral and take
the integral with respect to x because dx is on the outside.
Again, first with respect to y. The way we’re going to do that, we’ll go ahead and ignore
the outer integral for now but we just need to write it like this and we’ll draw brackets
to indicate that we’re focusing now on the integral with respect to y. So to take the
integral with respect to y term by term, we’ll start with 2x. The integral with respect to
y of 2x is going to be 2xy. And the reason is because we’re treating x as a constant
when we’re taking the integral with respect to y. Just as if you had the integral of 2
and you were taking it with respect to x, you’re integral would be 2x. Because this
is a constant, you just tack on the variable here. Similarly, if we’re treating x as
constant because we’re taking the integral with respect to y, then this whole term right
here, 2x is a constant. x is a constant. 2 is a constant. A constant times a constant
is going to be a constant. Because we’re taking it with respect to y, we just tack
on this y variable right here. So the variable here is 2xy. The integral of -7y is pretty
standard because we’re taking the integral with respect to y so that’s normal. We just
add 1 to the exponent to get a y^2 and we divide by our new exponent so 2 goes here
in our denominator. Then we’re going to be evaluating this integral
on the range 1 to 3 because those are our inside limits of integration. And then we
have to take our dx out here because this dx notation still goes with this integral
over here. We just keep those on the outside. So we’re going to simplify everything that
we got here by plugging in the limits of integration. And remember, we just took the integral with
respect to y so we’re going to be plugging in for y and leaving x as it is. So we get
2x(3) – 7/2 (3)^2 and then we’ll subtract from that what we get when we plug in 1 so
2x(1) – 7/2 (1)^2 and then dx here on the outside. Now if we simplify further, the integral
from -1 to 2 stays and we’ll get 6x – 63/2 – 2x + 7/2 and then we keep the dx here
on the outside. So we can simplify one more time by combining terms. 6x – 2x = 4x and
then -63/2 + 7/2 = -56/2. The -56/2 will simplify to -28 dx. So that’s about as simple as
we can get our inner integral. We could factor out a 4 here and get 4(x - 7) but it doesn’t
really buy us a whole lot so I’m not going to worry about it right now.
Let’s go ahead and now take our integral with respect to x. We finished with y and
now we’re taking it with respect to x because dx is the notation we’ve got left here.
So to take the integral of 4x – 28 with respect to x, pretty simply. Again, it’s
just going to be 4/2 x^2 – 28x and we’ll be evaluating that on the range -1 to 2. But
let’s simplify this first, we’ll get 4/2 x^2 – 28x = 2x^2 – 28x and we’re evaluating
that on -1 to 2. Doing so, we’ll plug in 2 first. We’ll get [2(2)^2 – 28(2)] and
then we subtract from that what we get when we plug in -1. 2(-1)^2 – 28(-1). 8 – 56
– 2 -28. So let’s see here. -56 - -28 will be a -76. -84 – 2 = -86. Plus 8 will
be -78. So -78 is our final answer for this double integral.
I hope this video helped you guys and I will see you in the next one. Bye!