Double Integrals

Volume under a surface, one direction at a time.

12 problems · about 24 minutes · F-IF.B.6

What this lesson teaches

The student sets up and evaluates double integrals over rectangular regions.

A double integral sums thin columns to find volume under a surface.
It is evaluated as two ordinary integrals, inner first, with the outer variable held constant.
Integrating the constant 1 returns the area of the region.

Straightforward practice. Get the method working first.

5 problems

A constant height of 4 over a rectangle 3 by 3. What is the double integral?
Answer 36
Why 36.
What does a double integral compute?
Answer The volume between a region of the plane and the surface above it.
Why Volume.
A constant height of 2 over a rectangle 5 by 6. What is the double integral?
Answer 60
Why 2 × 30.
The inner integral of a double integral treats the outer variable how? 1 as a constant, 2 as a variable.
Answer 1
Why The same way a partial derivative does.
Integrating 1 over a rectangle 7 by 4. What is the result?
Answer 28
Why That is just the area.

The same ideas with more to keep track of.

3 problems

A double integral of 90 over a region of area 15. What is the average value?
Answer 6
Why Divide by the area.
Does swapping the order of integration change the answer over a rectangle? 1 yes, 0 no.
Answer 0
Why Fubini's theorem.
A constant height of 3 over a rectangle 2 by 2. What is the double integral?
Answer 12
Why 3 × 4.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Order the steps of evaluating an iterated integral.
Answer 1. Integrate the inner variable, holding the outer one fixed 2. Substitute the inner limits 3. Integrate the resulting expression in the outer variable 4. Substitute the outer limits
Why Work from the inside outward.
A double integral of 0 over a region where the function is 0 everywhere. What is the volume?
Answer 0
Why No height means no volume.
The Flat Slab: A constant height of 5 over a rectangle 4 by 3. What is the volume?
Answer 60
Why 60.
The Area: Integrating the constant 1 over a region gives 24. What is the area of the region?
Answer 24
Why 24.