Exercise 520Calculus › Multivariable calculus › Multiple integralsHigh schoolAdded 3 Nov 2026

Double integral under a parabola

#double integral#type-I domain#parabola

Statement

Compute $\displaystyle I=\iint_D x^2\,dx\,dy$ where $D=\{(x,y):x\le1,\ y\ge0,\ y^2\le x\}$ .

Hint

For $x\in[0,1]$ , $y$ ranges from $0$ to $\sqrt x$ .

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Iterated. $I=\int_0^1\int_0^{\sqrt x}x^2\,dy\,dx$ .

Inner. $x^2\sqrt x$ .

Outer. $I=\int_0^1 x^{5/2}\,dx=\frac{2}{7}$ .

Conclusion. $I=2/7$ .