The area between two curves is the integral of the upper function minus the lower one.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Where the region ends
Usually at the points where the curves cross. Set them equal and solve.
When they swap places
If the curves cross inside the interval, split the integral there. Otherwise part of the area subtracts.
Area is positive
Getting a negative answer means the functions were the wrong way round, not that the area is negative.
Top minus bottom
The area between two curves is the integral of the upper minus the lower over the interval where they bound the region. The subtraction goes inside the integral, not after it.
Find the intersections and sketch
The limits are usually where the curves meet, found by setting them equal. Sketching first is what tells you which curve is on top, and skipping the sketch is how the subtraction ends up reversed.
Split where they cross
If the curves swap places inside the interval, split the integral at the crossing and take top minus bottom on each part separately. A single integral across a crossing subtracts the wrong way on one piece.
Sometimes integrate with respect to y
For regions bounded left and right, integrating in y — right minus left — is simpler and often avoids splitting. Choosing the variable to suit the region is a genuine decision worth making deliberately.
Step 2: Try It Yourself
Tap and try it out.
- Point(0, 0)
- Second point(2, 4)
- 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 Nadia Find the Region
Nadia finds the area between y = x and y = x² from 0 to 1.
- Step 1
On (0, 1) the line sits above the parabola — she checks at x = 0.5.
Step 4: Your Turn
Practice makes it stick.
The Crossings
Problem 1 of 2
y = x and y = x² cross at x = 0 and at which other x?
Which Is On Top
Problem 2 of 2
Between 0 and 1, which is greater at x = 0.5? 1 for x, 2 for x².
Between the Curves
1 of 8
y = 4 and y = 0 from 0 to 3. What is the area?
2 of 8
y = x and y = 0 from 0 to 4. What is the area?
3 of 8
y = 6 and y = 2 from 0 to 5. What is the area?
4 of 8
A negative answer for an area means what? 1 for the order was wrong, 2 for the area is negative.
5 of 8
y = x² and y = 0 from 0 to 3. What is the area?
6 of 8
If two curves cross inside the interval, must you split the integral? 1 for yes, 0 for no.
7 of 8
y = 2x and y = 0 from 0 to 3. What is the area?
8 of 8
y = 10 and y = 4 from 1 to 3. What is the area?
Step 5: Quick Check
Show what you know.
Question 1 of 1
y = 5 and y = 1 from 0 to 4. What is the area?
What You Learned
- The area between curves is the integral of the top function minus the bottom one.
- The limits are usually where the curves cross.
- Split the integral wherever they swap places, and expect a positive answer.