Solve ∫ (from 0 to frac{ 1) of {4}}x^2y1/3 wrt y

$$\int_{ 0 }^{ \frac{ 1 }{ 4 } } { x }^{ 2 } y \frac{ 1 }{ 3 } d y$$

Evaluate

$$\frac{x^{2}}{96}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \frac{x^{2}y}{3}\mathrm{d}y$$

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

$$\frac{x^{2}}{3}\int y\mathrm{d}y$$

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}.

$$\frac{x^{2}}{3}\times \frac{y^{2}}{2}$$

Simplify.

$$\frac{x^{2}y^{2}}{6}$$

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.

$$\frac{1}{6}x^{2}\times \left(\frac{1}{4}\right)^{2}-\frac{1}{6}x^{2}\times 0^{2}$$

Simplify.

$$\frac{x^{2}}{96}$$

Differentiate w.r.t. x

$$\frac{x}{48}$$