Exercise · ZW-AREA-BETWEEN-CURVES-006 · ★☆☆ Basic
If \(f\ge g\) on \([a,b]\), which integral gives the area between \(f\) and \(g\) on that interval?
Choose one option.
Show answer
- A. \(\int_a^b [f(x)-g(x)]\,dx\)
Explanation
Since \(f\ge g\), the difference \(f-g\) is nonnegative, and its integral over \([a,b]\) is the area. The integral of \(g-f\) would give a negative value.
Review the concept: Area between curves
The area between the graphs of \(f\) and \(g\) on \([a,b]\), with \(f\ge g\), is the integral of the difference \(f-g\). If the curves cross, the interval is split at the crossing points.
\[A=\int_a^b\,[f(x)-g(x)]\,dx\]
More practice problems: area between curves
- ZW-AREA-BETWEEN-CURVES-001 Find the area between y=x and y=x^2 on [0,1].
- ZW-AREA-BETWEEN-CURVES-002 Find the area between y=2 x and y=x^2 on [0,2].
- ZW-AREA-BETWEEN-CURVES-003 Find the area between y=√x and y=x^2 on [0,1].
- ZW-AREA-BETWEEN-CURVES-004 Find the area between y=x + 2 and y=x^2 on [-1,2].
- ZW-AREA-BETWEEN-CURVES-005 Find the area between y=x^3 and y=0 on [0,1].
- ZW-AREA-BETWEEN-CURVES-007 Find the area between y=sin x and y=0 on [0,π].
- ZW-AREA-BETWEEN-CURVES-008 Find the area between y=x and y=x^3 on [0,1].
- ZW-AREA-BETWEEN-CURVES-009 Find the area between y=x^2 and y=0 on [1,2].