Solve ∫ (from 0 to 2) of (-6x^2) wrt x

$$\int_{ 0 }^{ 2 } (-6 { x }^{ 2 } ) d x$$

Evaluate

$$-16$$

Solution Steps

Evaluate the indefinite integral first.

$$\int -6x^{2}\mathrm{d}x$$

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

$$-6\int x^{2}\mathrm{d}x$$

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 -6 times \frac{x^{3}}{3}.

$$-2x^{3}$$

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.

$$-2\times 2^{3}+2\times 0^{3}$$

Simplify.

$$-16$$