Solve ∫ (from 0 to 2 pi) of xy wrt xdy

$$\int _ { 0 } ^ { 2 \pi } x y d x d y$$

Evaluate

$$2d\left(\pi x\right)^{2}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int xydx\mathrm{d}y$$

Factor out the constant using \int af\left(y\right)\mathrm{d}y=a\int f\left(y\right)\mathrm{d}y.

$$xdx\int y\mathrm{d}y$$

Since \int y^{k}\mathrm{d}y=\frac{y^{k+1}}{k+1} for k\neq -1, replace \int y\mathrm{d}y with \frac{y^{2}}{2}.

$$xdx\times \frac{y^{2}}{2}$$

Simplify.

$$\frac{x^{2}dy^{2}}{2}$$

The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.

$$\frac{1}{2}x^{2}d\times \left(2\pi \right)^{2}-\frac{1}{2}x^{2}d\times 0^{2}$$

Simplify.

$$2x^{2}d\pi ^{2}$$

Differentiate w.r.t. x

$$4\pi ^{2}dx$$