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

$$\int_{ -1 }^{ 1 } 1- { y }^{ 2 } - { y }^{ 2 } -1 d y$$

Evaluate

$$-\frac{4}{3}\approx -1.333333333$$

Solution Steps

Combine -y^{2} and -y^{2} to get -2y^{2}.

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

Subtract 1 from 1 to get 0.

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

Evaluate the indefinite integral first.

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

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

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

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}.

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

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.

$$-\frac{2}{3}\times 1^{3}+\frac{2}{3}\left(-1\right)^{3}$$

Simplify.

$$-\frac{4}{3}$$