Solve ∫ (from - 2 to 2) of (14y^2-56) wrt y

$$\int _ { - 2 } ^ { 2 } ( 14 y ^ { 2 } - 56 ) d y$$

Evaluate

$$-\frac{448}{3}\approx -149.333333333$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 14y^{2}-56\mathrm{d}y$$

Integrate the sum term by term.

$$\int 14y^{2}\mathrm{d}y+\int -56\mathrm{d}y$$

Factor out the constant in each of the terms.

$$14\int y^{2}\mathrm{d}y+\int -56\mathrm{d}y$$

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 14 times \frac{y^{3}}{3}.

$$\frac{14y^{3}}{3}+\int -56\mathrm{d}y$$

Find the integral of -56 using the table of common integrals rule \int a\mathrm{d}y=ay.

$$\frac{14y^{3}}{3}-56y$$

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.

$$\frac{14}{3}\times 2^{3}-56\times 2-\left(\frac{14}{3}\left(-2\right)^{3}-56\left(-2\right)\right)$$

Simplify.

$$-\frac{448}{3}$$