Solve ∫ (from 0 to 2) of y^2x wrt y

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

Evaluate

$$\frac{8x}{3}$$

Solution Steps

Evaluate the indefinite integral first.

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

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

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

Since \int y^{k}\mathrm{d}y=\frac{y^{k+1}}{k+1} for k\neq -1, replace \int y^{2}\mathrm{d}y with \frac{y^{3}}{3}.

$$x\times \frac{y^{3}}{3}$$

Simplify.

$$\frac{xy^{3}}{3}$$

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.

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

Simplify.

$$\frac{8x}{3}$$

Differentiate w.r.t. x

$$\frac{8}{3} = 2\frac{2}{3} = 2.6666666666666665$$