Area Under and Between Curves
The area bounded by curves is the definite integral of the difference of the upper and lower functions across their intersection range.
What is Area Under and Between Curves?
The area bounded by curves is the definite integral of the difference of the upper and lower functions across their intersection range.
Key formula / rule: Find the intersection points to set the limits of integration.
Key points
- Derive the area of a region bounded by two curves.
- Model a bounded planar region as one or more definite integrals.
Common exam trap
Reporting a negative value as an area.
Definitions
- Term
Area Under and Between Curves
- Meaning
The area bounded by curves is the definite integral of the difference of the upper and lower functions across their intersection range.
- Term
Area Under and Between Curves — explanation
- Meaning
Sketching first decides which curve is on top and where the region begins and ends. Without the sketch the sign and the limits are guesswork.
Learning objectives
Derive the area of a region bounded by two curves.
Model a bounded planar region as one or more definite integrals.
Formulae
- Key point
Find the intersection points to set the limits of integration.
- Key point
Integrate upper minus lower, or right minus left for horizontal strips.
- Key point
Split the integral where the curves cross inside the range.
Prerequisites
MAT-U2-IC-T1-S2-C1
Common mistakes
Reporting a negative value as an area.
Using a single integral where the upper curve changes midway.
Keywords
Area
Under
Between
Curves
Practice preview
Find the area bounded by the curve y = x^2, the x-axis, and the lines x = 0 and x = 2.…
easy
Determine the area of the region bounded by the curves y = x and y = x^2.…
easy
Calculate the area of the region bounded by the curve y = |x|, the x-axis, and the line y = 1.…
medium