Solve ∫ (from 0 to 1 - y) of x^2 wrt x

$$\int _ { 0 } ^ { 1 - y } x ^ { 2 } d x$$

Evaluate

$$\frac{\left(1-y\right)^{3}}{3}$$

Solution Steps

Evaluate the indefinite integral first.

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

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

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

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.

$$\frac{1}{3}\left(1-y\right)^{3}-\frac{0^{3}}{3}$$

Simplify.

$$\frac{\left(1-y\right)^{3}}{3}$$

Differentiate w.r.t. y

$$-\left(1-y\right)^{2}$$