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

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

Evaluate

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

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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}. Multiply -1 times \frac{y^{3}}{3}.

$$-\frac{y^{3}}{3}+\int 1\mathrm{d}y$$

Find the integral of 1 using the table of common integrals rule \int a\mathrm{d}y=ay.

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

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.

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

Simplify.

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