► My Multiple Integrals course: https://www.kristakingmath.com/multiple-integrals-course
Skip to section:
0:15 // Recap of what the double integral represents
1:22 // The triple integral has two uses (volume and mass)
1:45 // How to use the triple integral to find volume
8:59 // Why the triple integral does more than the double integral
11:19 // How to use the triple integral to find mass, when the volume has variable density
It can be difficult to visualize what a triple integral represents, which is why in this video we’ll be answering the question, “What am I finding when I evaluate a triple integral?”
In order to answer this question, we’ll compare the triple integral to a double integral, so that we understand exactly how to transition from double integrals into triple integrals. Every piece of the double integral, like the integral, the bounds or limits of integration, the function which is the integrand, and the differential (usually dydx) will all translate into a corresponding piece of the triple integral.
The interesting thing about the triple integral is that it can be used in two ways. In contrast, single integrals only find area under the curve and double integrals only find volume under the surface. But triple integrals can be used to 1) find volume, just like the double integral, and to 2) find mass, when the volume of the region we’re interested in has variable density.
In this way, triple integrals let us do more than we were able to do with double integrals. We’re able to add in the extra dimension of variable density inside the volume, and based on that variable density, find the mass of the volume, as opposed to only being able to find volume, which is what we were limited to in the double integral.
If we want to describe double and triple integrals with words, we can say that for the double integral, we’re integrating a multivariable function f(x,y) over the region R which is defined for x on the interval [a,b] and for y on the interval [c,d], using vertical slices of volume, in order to find the total volume under the surface f(x,y) but above the xy-plane.
In contrast, we can say that for the triple integral, we’re integrating a multivariable function for density f(x,y,z) for the volume B which is defined for x on the interval [a,b] and for y on the interval [c,d] and for z on the interval [r,s], by slicing the volume in three direction to get tiny pieces (or boxes) of volume, in order to find the total mass of the volume.
Music by Joakim Karud: http://soundcloud.com/joakimkarud
● ● ● GET EXTRA HELP ● ● ●
If you could use some extra help with your math class, then check out my website // http://www.kristakingmath.com
● ● ● CONNECT WITH KRISTA ● ● ●
Hi, I’m Krista! I make math courses to keep you from banging your head against the wall. ;)
Math class was always so frustrating for me. I’d go to a class, spend hours on homework, and three days later have an “Ah-ha!” moment about how the problems worked that could have slashed my homework time in half. I’d think, “WHY didn’t my teacher just tell me this in the first place?!”
So I started tutoring to keep other people out of the same aggravating, time-sucking cycle. Since then, I’ve recorded tons of videos and written out cheat-sheet style notes and formula sheets to help every math student—from basic middle school classes to advanced college calculus—figure out what’s going on, understand the important concepts, and pass their classes, once and for all. Interested in getting help? Learn more here: http://www.kristakingmath.com
FACEBOOK // https://www.facebook.com/kristakingmath/
INSTAGRAM // https://www.instagram.com/kristakingmath/
Tagged under: triple integrals, triple integrals work, sketch triple integrals, visualize triple integrals, draw triple integrals, triple integrals, triple integrals, triple integrals, triple integrals, triple integrals ,mass,volume,box ,multiple integrals,calculus 3,calc 3,multivariable,multivariate,multivariable calculus,multivariate calculus,calculus
Clip makes it super easy to turn any public video into a formative assessment activity in your classroom.
Add multiple choice quizzes, questions and browse hundreds of approved, video lesson ideas for Clip
Make YouTube one of your teaching aids - Works perfectly with lesson micro-teaching plans
1. Students enter a simple code
2. You play the video
3. The students comment
4. You review and reflect
* Whiteboard required for teacher-paced activities
With four apps, each designed around existing classroom activities, Spiral gives you the power to do formative assessment with anything you teach.
Carry out a quickfire formative assessment to see what the whole class is thinking
Create interactive presentations to spark creativity in class
Student teams can create and share collaborative presentations from linked devices
Turn any public video into a live chat with questions and quizzes