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.
Finding the limits
The curves usually bound the region themselves. Set them equal and solve to find where they cross.
Which is on top
Test one x-value inside the interval. The larger output belongs to the upper curve.
When they trade places
If the curves cross inside the region, split the integral at the crossing and take top minus bottom on each piece.
The axis needs no special case
Area under a curve is just the area between it and y = 0, so this rule already contains the earlier one.
Accumulation in context
If two curves are rates, such as flow in and flow out, the area between them is the net accumulated amount.
Top minus bottom
The area between two curves is the integral of the upper function minus the lower, over the interval where they bound the region. The subtraction happens inside the integral, not afterwards.
Find the intersections first
The limits of integration are usually where the curves cross, found by setting them equal. Sketching the region before integrating is what tells you which curve is on top and where the region actually is.
When they swap places
If the curves cross inside the interval, split the integral at the crossing and take top minus bottom separately on each piece. A single integral across a crossing subtracts the wrong way on one part.
Sometimes integrate with respect to y
For regions bounded left and right rather than above and below, integrating in y is far simpler — right minus left, with respect to y. Choosing the variable to suit the region often avoids splitting entirely.
Step 2: Try It Yourself
Tap and try it out.
- Point(-1, 1)
- Second point(2, 4)
- Slope between them1
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 Kofi Find an Area Between Curves
Kofi needs the area between y = x and y = x² from 0 to 1.
- Step 1
Setting them equal gives x = x², so they cross at x = 0 and x = 1.
Step 4: Your Turn
Practice makes it stick.
The Crossings
Problem 1 of 2
y = x and y = x² cross at x = 0 and which other x?
The Tank
Problem 2 of 2
Water flows in at 8 L/min and out at 3 L/min for 10 minutes. What is the net gain, in litres?
L
Between the Curves
1 of 8
y = 4 and y = 1, from x = 0 to x = 5. What is the area between them?
2 of 8
y = x² and y = 0 from 0 to 3. Antiderivative x³/3. Area?
3 of 8
y = x and y = x² cross at x = 0 and x = 1. On (0, 1), which is on top? 1 the line, 2 the parabola.
4 of 8
Flow in 10 L/min, out 4 L/min, for 6 minutes. Net gain in litres?
5 of 8
y = 6 and y = 2 from x = 1 to x = 4. Area between?
6 of 8
If the curves cross inside the region, how many integrals are needed at minimum?
7 of 8
Put the area-between-curves method in order.
- 1Test a point inside to see which is on top.
- 2Integrate top minus bottom between the crossings.
- 3Split the integral if the curves trade places.
- 4Set the two functions equal to find the crossings.
8 of 8
y = 9 and y = 4 from x = 0 to x = 2. Area between?
Step 5: Quick Check
Show what you know.
Question 1 of 2
y = 7 and y = 2 from x = 0 to x = 4. What is the area between them?
Question 2 of 2
What is the integrand for an area between curves?
What You Learned
- The area between curves is the integral of top minus bottom.
- Set the functions equal to find the limits, then test a point to see which is on top.
- When two rates are graphed, the area between them is the net accumulation.