Solve ∫ (from 0 to 2) of 24x^2 wrt x

$$\int _ { 0 } ^ { 2 } 24 x ^ { 2 } d x$$

Evaluate

$$64$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 24x^{2}\mathrm{d}x$$

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

$$24\int x^{2}\mathrm{d}x$$

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 24 times \frac{x^{3}}{3}.

$$8x^{3}$$

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.

$$8\times 2^{3}-8\times 0^{3}$$

Simplify.

$$64$$