Solve ∫ (from 20 to 140) of 20y^2 wrt y

$$\int _ { 20 } ^ { 140 } 20 y ^ { 2 } d y$$

Evaluate

$$18240000$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 20y^{2}\mathrm{d}y$$

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

$$20\int y^{2}\mathrm{d}y$$

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}.

$$\frac{20y^{3}}{3}$$

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.

$$\frac{20}{3}\times 140^{3}-\frac{20}{3}\times 20^{3}$$

Simplify.

$$18240000$$