Solve ∫ (from 0 to 4) of 5t-t^2dt

$$\int _ { 0 } ^ { 4 } 5 t - t ^ { 2 } d t$$

Evaluate

$$\frac{56}{3}\approx 18.666666667$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 5t-t^{2}\mathrm{d}t$$

Integrate the sum term by term.

$$\int 5t\mathrm{d}t+\int -t^{2}\mathrm{d}t$$

Factor out the constant in each of the terms.

$$5\int t\mathrm{d}t-\int t^{2}\mathrm{d}t$$

Since \int t^{k}\mathrm{d}t=\frac{t^{k+1}}{k+1} for k\neq -1, replace \int t\mathrm{d}t with \frac{t^{2}}{2}. Multiply 5 times \frac{t^{2}}{2}.

$$\frac{5t^{2}}{2}-\int t^{2}\mathrm{d}t$$

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}. Multiply -1 times \frac{t^{3}}{3}.

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

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.

$$\frac{5}{2}\times 4^{2}-\frac{4^{3}}{3}-\left(\frac{5}{2}\times 0^{2}-\frac{0^{3}}{3}\right)$$

Simplify.

$$\frac{56}{3}$$