A single integral summed thin rectangles to find area under a curve. A double integral sums thin columns to find volume under a surface.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Iterated integrals
You never integrate in two variables at once. Integrate in x holding y fixed, then integrate the result in y.
The inner integral
The inner integral treats the outer variable as a constant, exactly the way a partial derivative does.
Fubini's theorem
Over a rectangle with a well-behaved function, the order of integration does not change the answer. It can change the difficulty enormously.
Integrating 1
Integrating the constant 1 over a region gives the area of that region. A column of height 1 has volume equal to its base.
Average value
The double integral divided by the area of the region is the average value of the function over that region.
Volume under a surface
A double integral sums f(x, y) times tiny areas over a region, giving the volume beneath the surface. It is the two-variable analogue of area under a curve.
Computed one variable at a time
Integrate with respect to one variable holding the other fixed, then integrate the result. Each step is an ordinary single-variable integral, which is what makes the computation tractable.
The limits describe the region
The inner limits may depend on the outer variable, which is how non-rectangular regions are handled. Sketching the region before writing limits is essential and is where most errors originate.
Changing the order
For a well-behaved function the order can be swapped, but the limits must be rewritten from the region, not merely exchanged. Sometimes one order is elementary and the other is intractable.
Step 2: Try It Yourself
Tap and try it out.
- Point(0, 1)
- Second point(2, 5)
- Slope between them2
Slide the second point towards the first. The slope between them approaches the slope of the curve at that point.
Step 3: Watch an Example
One step at a time.
Watch Bilal Integrate Twice
Bilal integrates f(x, y) = 6 over the rectangle from x = 0 to 2 and y = 0 to 3.
- Step 1
He does the inner integral in x first, treating y as a constant.
Step 4: Your Turn
Practice makes it stick.
The Flat Slab
Problem 1 of 2
A constant height of 5 over a rectangle 4 by 3. What is the volume?
The Area
Problem 2 of 2
Integrating the constant 1 over a region gives 24. What is the area of the region?
Integrate Twice
1 of 8
A constant height of 2 over a rectangle 5 by 6. What is the double integral?
2 of 8
The inner integral of a double integral treats the outer variable how? 1 as a constant, 2 as a variable.
3 of 8
Integrating 1 over a rectangle 7 by 4. What is the result?
4 of 8
A double integral of 90 over a region of area 15. What is the average value?
5 of 8
Does swapping the order of integration change the answer over a rectangle? 1 yes, 0 no.
6 of 8
A constant height of 3 over a rectangle 2 by 2. What is the double integral?
7 of 8
Order the steps of evaluating an iterated integral.
- 1Substitute the inner limits
- 2Integrate the resulting expression in the outer variable
- 3Substitute the outer limits
- 4Integrate the inner variable, holding the outer one fixed
8 of 8
A double integral of 0 over a region where the function is 0 everywhere. What is the volume?
Step 5: Quick Check
Show what you know.
Question 1 of 2
A constant height of 4 over a rectangle 3 by 3. What is the double integral?
Question 2 of 2
What does a double integral compute?
What You Learned
- 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.