Solve ∫ (from 0 to 2) of 4t^2dt

$$\int _ { 0 } ^ { 2 } 4 t ^ { 2 } d t$$

Evaluate

$$\frac{32}{3}\approx 10.666666667$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 4t^{2}\mathrm{d}t$$

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

$$4\int t^{2}\mathrm{d}t$$

Since \int t^{k}\mathrm{d}t=\frac{t^{k+1}}{k+1} for k\neq -1, replace \int t^{2}\mathrm{d}t with \frac{t^{3}}{3}.

$$\frac{4t^{3}}{3}$$

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.

$$\frac{4}{3}\times 2^{3}-\frac{4}{3}\times 0^{3}$$

Simplify.

$$\frac{32}{3}$$