Solve ∫ (from 0 to 3) of x^2y wrt y

$$\int _ { 0 } ^ { 3 } x ^ { 2 } y d y$$

Evaluate

$$\frac{9x^{2}}{2}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int x^{2}y\mathrm{d}y$$

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

$$x^{2}\int y\mathrm{d}y$$

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}.

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

Simplify.

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

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.

$$\frac{1}{2}x^{2}\times 3^{2}-\frac{1}{2}x^{2}\times 0^{2}$$

Simplify.

$$\frac{9x^{2}}{2}$$

Differentiate w.r.t. x

$$9x$$