Solve ∫ (from 0 to 1) of y^2e^2 wrt y

$$\int_{ 0 }^{ 1 } { y }^{ 2 } { e }^{ 2 } d y$$

Evaluate

$$\frac{e^{2}}{3}\approx 2.4630187$$

Solution Steps

Evaluate the indefinite integral first.

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

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

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

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}.

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

Simplify.

$$\frac{e^{2}y^{3}}{3}$$

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.

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

Simplify.

$$\frac{e^{2}}{3}$$