Video Transcript
Evaluate the triple integral.
We use a triple integral to
integrate over a three-dimensional region. And whilst it can look pretty
intimidating, the process itself is fairly straightforward. We’ll begin by integrating 𝑧 with
respect to 𝑧 within the limits of one and zero. We’ll then integrate 𝑦 with
respect to 𝑦 between two and zero. And finally, we’ll integrate 𝑥
with respect to 𝑥 between three and zero.
So let’s begin with then
integrating 𝑧 with respect to 𝑧. The integral of 𝑧 with respect to
𝑧 is 𝑧 squared over two. And with no limits, we have that
constant of integration. Remember, to integrate, we add one
to the exponent. And then, we divide by the value of
this new exponent. Since we’re integrating between the
limits one and zero, we’re going to need to substitute these values into our
expression and find the difference. That’s one squared divided by two
minus zero squared divided by two, which is one-half. And we’re now left with a double
integral.
We’re now going to integrate 𝑦
with respect to 𝑦 between the limits two and zero. The integral of 𝑦 with respect to
𝑦 is 𝑦 squared divided by two. And between our limits, it’s two
squared divided by two minus zero squared divided by two, which is two. And we’re left with one simple
integral. We have the integral of two times
one-half 𝑥 with respect to 𝑥 between three and zero. Now actually, two multiplied by
one-half is one. So we’re simply integrating 𝑥 with
respect to 𝑥.
The integral of 𝑥 with respect to
𝑥 is 𝑥 squared divided by two. So we need to evaluate these
between the limits of three and zero. That’s three squared divided by two
minus zero squared divided by two, which is simply 4.5. And we’ve evaluated our triple
integral. It’s 4.5.
Now, it’s important to realise that
whilst we picked a specific order, that is, we integrated 𝑧 with respect to 𝑧 and
then 𝑦 with respect to 𝑦 and 𝑥 with respect to 𝑥, we could have chosen any other
order as long as we ensured that each integral was evaluated between the limits
given.
