Double Integrals

Volume under a surface, one direction at a time.

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

NameDate

Figure — use these to answer the problems

The shaded area is what a single integral finds. A double integral does this once for every slice of a surface.

y = 1x² + 0x + 1

lower limit0

upper limit2

Figure 1

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
2
What does a double integral compute?
- The volume between a region of the plane and the surface above it.
- The slope of the surface.
A constant height of 2 over a rectangle 5 by 6. What is the double integral?
Answer
The inner integral of a double integral treats the outer variable how? 1 as a constant, 2 as a variable.
Answer
Integrating 1 over a rectangle 7 by 4. What is the result?
Answer

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
Does swapping the order of integration change the answer over a rectangle? 1 yes, 0 no.
Answer
A constant height of 3 over a rectangle 2 by 2. What is the double integral?
Answer

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

4 problems

9
Order the steps of evaluating an iterated integral.
Write 1 to 4 in the boxes to put these in order.
- Integrate the inner variable, holding the outer one fixed
- Substitute the inner limits
- Integrate the resulting expression in the outer variable
- Substitute the outer limits
A double integral of 0 over a region where the function is 0 everywhere. What is the volume?
Answer
The Flat Slab
A constant height of 5 over a rectangle 4 by 3. What is the volume?
Answer
The Area
Integrating the constant 1 over a region gives 24. What is the area of the region?
Answer