Area between curves
Integrate the top curve minus the bottom curve over the interval where they enclose a region. Find the crossing points first.
Last updated
Analogy
Cutting a strip of pastry between two lines: stack up all the thin vertical slices.
Example
They meet at x = 0 and x = 1.On (0, 1), x is above x^2.Integral of (x - x^2) = 1/2 - 1/3.Area = 1/6.
In practice
Used for areas, and later for volumes and probabilities.
Practice
Flashcards
- What is the area between y = x and y = x^2 on [0, 1]?1/6.
Interview questions
- Find the area enclosed between y = x and y = x^2.
- Solve x = x^2 for crossing points x = 0 and 1
- Line is above parabola on (0, 1)
- Integrate x - x^2 from 0 to 1
- Result 1/2 - 1/3 = 1/6
Questions people ask
How do I know which curve is on top?
Pick a test value between the crossing points and compare the two functions there. At x = 0.5, y = x gives 0.5 while y = x squared gives 0.25, so the line is on top. Swapping them would give a negative area.
What if the curves cross more than once?
Split the integral at every crossing point. On each piece, subtract the lower function from the upper one. Then add the pieces. If you use one formula across a crossing, areas above and below cancel and you get the wrong total.