← Unit 8: Applications of Integration
Area Between Curves
The area between two curves is found by integrating the difference between the top (or right) function and the bottom (or left) function over the interval where they bound a region.
Setting up the integral
For vertical strips: Area = ∫[a,b] (top function - bottom function) dx, where a and b are the x-coordinates of the intersection points. Always find intersection points first by setting the two functions equal.
If the curves cross within the interval (top/bottom switch roles), split the integral at each crossing point and use the correct order (top - bottom) on each piece.
Worked Example
Find the area between y=x² and y=x on [0,1].
- On [0,1], x ≥ x² (check: at x=0.5, 0.5 > 0.25), so x is on top.
- Area = ∫[0,1] (x - x²) dx = [x²/2 - x³/3] from 0 to 1
- = (1/2 - 1/3) - 0 = 1/6
Answer: 1/6
Key terms (1)
- Area between curves
- ∫[a,b] (top - bottom) dx, integrated between the curves' intersection points.
Practice Quiz
Question 1 of 2Score so far: 0/0
To find the area between y=f(x) and y=g(x), why must you find their intersection points first?