Exercise 1519Calculus › Multivariable calculus › Vector calculusL3Added 2 Nov 2026

Green-Riemann formula

#Green-Riemann#line integral#double integral

Statement

Let $C$ be the unit circle oriented counterclockwise. Compute
$\int_C (x^2 - y)dx + (y^2 + x)dy$
using the Green-Riemann formula.

Hint

Apply Green-Riemann: $\int_C Pdx + Qdy = \iint_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dx\,dy$ .

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Solution

Identification. $P(x,y) = x^2 - y$ , $Q(x,y) = y^2 + x$ .

Partial derivatives.
$\frac{\partial Q}{\partial x} = 1, \qquad \frac{\partial P}{\partial y} = -1.$
Hence
$\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 1 - (-1) = 2.$

Green-Riemann.
$\int_C Pdx + Qdy = \iint_D 2 \, dx\,dy = 2 \cdot \text{Area}(D) = 2\pi.$

Remark. One can also compute directly by parametrizing the circle, but Green-Riemann avoids trigonometric computation.

Green-Riemann formula

Statement

Solution

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