Triple Integrals¶
The idea of double integrals over a plane region generalizes to $n$-tuple integrals over regions in $n$-dimensional space. We're not going to go beyond $3$ dimensions in this class, but you may need to in your careers.
Let's study the $3$-dimensional case now!
Triple Integrals Over Boxes¶
We begin with the simplest case of a triple integral over a $3$-dimensional region known as a "right rectangular prism," or more succinctly, a box.
Video Lecture
To compute triple integrals, we will generally not want to have to use the Riemann sum definition unless we have to. (Sometimes we will have to use it, so don't forget it exists, or that it's the definition.)
Luckily, a few of our favorite theorems will usually apply. So we will take advantage of them.
Video Lecture
1. Example¶
Compute $\displaystyle \int\!\!\!\int\!\!\!\int_B xyz^2\, dV$, where $B = [0,1]\times [-1,2] \times [0,3]$.
Check your answer
$$\frac{27}{4}$$
Video Solution
Triple Integrals Over General Regions¶
Just as we did with double integrals we can generalize the idea of a triple integral to make sense over a $3$-dimensional region bounded between two or more surfaces.
Video Lecture
2. Example¶
Find the volume of the solid region that lies inside the cylinder $x^2 + y^2 = 4$, above the plane $z = 0$, and below the paraboloid $z = 5 - x^2 - y^2$.
Hint
Sketch the region, then use the fact that $\text{Vol}(E) = \int\!\!\!\int\!\!\!\int_E 1\, dV$ to write down a triple integral. When it comes time to integrate, think about changing coordinates after the first iterated integral.
Check your answer
$$12\pi$$
Video Solution
3. Example¶
Sketch and/or describe the solid whose volume is given by the iterated integral.
$$
\int_0^2 \int_0^{2-y} \int_0^{4-y^2} \ dx\, dz\, dy.
$$
Video Solution
Discussion¶
Questions? You can ask them here.