Areas Between Curves
Calculate the area enclosed between two curves by finding intersection points and setting up the correct integral.
Area Between Two Curves
If two curves y = f(x) and y = g(x) are continuous on [a, b] and f(x) ≥ g(x) throughout the interval, then the area between the curves is:
Area = ∫ab [f(x) − g(x)] dx
Always subtract the lower curve from the upper curve (top minus bottom).
Think of it this way: the area between two curves equals the area under the upper curve minus the area under the lower curve. This formula works even when both curves are below the x-axis.
Visual: Area Between Curves
y = f(x) (upper)
y = g(x) (lower)
Shaded Area
The area between the curves is the integral of (top − bottom) from a to b.
Finding Intersection Points
To find the limits of integration, set the two functions equal and solve for x. The intersection points become the bounds of the integral.
Example: y = x2 and y = 2x
Step 1: Set equal: x2 = 2x, so x2 − 2x = 0, giving x(x − 2) = 0.
Step 2: Intersections at x = 0 and x = 2.
Step 3: Check which is on top: at x = 1, f(1) = 2(1) = 2 and g(1) = 12 = 1. So y = 2x is above y = x2.
Tip: Always test a point between the intersections to determine which curve is on top. If the curves switch positions, you need to split the integral.
When the Curves Cross Each Other
If f(x) and g(x) switch positions (one is on top, then the other), you must split the integral at each crossing point. For each sub-interval, always put the upper curve first.
Area = ∫ac [f(x) − g(x)] dx + ∫cb [g(x) − f(x)] dx
where f is on top for [a, c] and g is on top for [c, b].
Alternatively, you can use the absolute value approach: Area = ∫ab |f(x) − g(x)| dx. In practice, you still need to split the integral to evaluate it.
Key Vocabulary
Intersection Point
A point where two curves meet, found by setting f(x) = g(x).
Upper Curve
The curve with the greater y-value at a given x — it goes on top in the subtraction.
Enclosed Region
The bounded area trapped between two curves from one intersection to another.
Top Minus Bottom
The key principle: always integrate f(x) − g(x) where f is above g.
Worked Examples
Find the area between y = x2 and y = 2x.
Step 1: Intersections: x2 = 2x gives x = 0 and x = 2. On [0,2], 2x ≥ x2.
Step 2: Area = ∫02 (2x − x2) dx = [x2 − x3/3]02
Step 3: = (4 − 8/3) − 0 = 4/3
Answer: Area = 4/3 square units
Find the area enclosed between y = x2 − 1 and y = 3 − x2.
Step 1: Set equal: x2 − 1 = 3 − x2, so 2x2 = 4, x2 = 2, x = ±√2.
Step 2: At x = 0: upper = 3 − 0 = 3, lower = 0 − 1 = −1. So (3 − x2) is on top.
Step 3: Area = ∫−√2√2 [(3 − x2) − (x2 − 1)] dx = ∫−√2√2 (4 − 2x2) dx
= [4x − 2x3/3]−√2√2 = 2(4√2 − 4√2/3) = 2 × 8√2/3
Answer: Area = 16√2/3 ≈ 7.54 square units
Find the area between y = x and y = x2 from x = 0 to x = 2.
Step 1: They intersect at x = 0 and x = 1 (from x = x2). For x in (0,1): x > x2. For x in (1,2): x2 > x.
Step 2: Split the integral:
A1 = ∫01 (x − x2) dx = [x2/2 − x3/3]01 = 1/2 − 1/3 = 1/6
A2 = ∫12 (x2 − x) dx = [x3/3 − x2/2]12 = (8/3 − 2) − (1/3 − 1/2) = 2/3 + 1/6 = 5/6
Answer: Total area = 1/6 + 5/6 = 1 square unit
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
What is the first step to finding the area between two curves?
Question 2
Find the area between y = 4 − x2 and y = 0 (the x-axis) for the enclosed region.
Question 3
In the formula ∫ab [f(x) − g(x)] dx, which function is f(x)?
Question 4
Find the area enclosed between y = x2 and y = x + 2.
Question 5
When two curves cross within the integration interval, you should:
Key Concepts Summary
- ● Area between curves = ∫ab [upper − lower] dx.
- ● Find intersection points by setting f(x) = g(x) to determine limits.
- ● Test a point between intersections to determine which curve is on top.
- ● If curves cross within the interval, split the integral at each crossing.
- ● This method works regardless of whether the curves are above or below the x-axis.