Solve ∫ (from 0 to 2) of 50/3*0.1t^4dt

$$\int _ { 0 } ^ { 2 } \frac { 50 } { 3 } \times 0.1 t ^ { 4 } d t$$

Evaluate

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

Solution Steps

Evaluate the indefinite integral first.

$$\int \frac{\frac{50}{3}t^{4}}{10}\mathrm{d}t$$

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

$$\frac{\frac{50}{3}}{10}\int t^{4}\mathrm{d}t$$

Since \int t^{k}\mathrm{d}t=\frac{t^{k+1}}{k+1} for k\neq -1, replace \int t^{4}\mathrm{d}t with \frac{t^{5}}{5}.

$$\frac{\frac{50}{3}}{10}\times \frac{t^{5}}{5}$$

Simplify.

$$\frac{t^{5}}{3}$$

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.

$$\frac{2^{5}}{3}-\frac{0^{5}}{3}$$

Simplify.

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