Exercise · ZW-AREA-BETWEEN-CURVES-008 · ★★☆ Intermediate
Find the area between \(y=x\) and \(y=x^{3}\) on \([0,1]\).
Choose one option.
Show answer
- C. \(\frac{1}{4}\)
Explanation
On \([0,1]\) we have \(x\ge x^{3}\), so the area is \(\int_0^1 (x-x^{3})\,dx=\frac{1}{2}-\frac{1}{4}=\frac{1}{4}\).
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-006 If f≥ g on [a,b], which integral gives the area between f and g on that interval?
- ZW-AREA-BETWEEN-CURVES-007 Find the area between y=sin x and y=0 on [0,π].
- ZW-AREA-BETWEEN-CURVES-009 Find the area between y=x^2 and y=0 on [1,2].