Solve ∫ (from - 1 to 3) of (-3x^2)

$$\int _ { - 1 } ^ { 3 } ( - 3 x ^ { 2 } )$$

Evaluate

$$-28$$

Solution Steps

Evaluate the indefinite integral first.

$$\int -3x^{2}\mathrm{d}x$$

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

$$-3\int x^{2}\mathrm{d}x$$

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}. Multiply -3 times \frac{x^{3}}{3}.

$$-x^{3}$$

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.

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

Simplify.

$$-28$$