Solve ∫ (from 1 to 2) of (45t^2-2t^3)dt

$$\int _ { 1 } ^ { 2 } ( 45 t ^ { 2 } - 2 t ^ { 3 } ) d t$$

Evaluate

$$\frac{195}{2}=97.5$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 45t^{2}-2t^{3}\mathrm{d}t$$

Integrate the sum term by term.

$$\int 45t^{2}\mathrm{d}t+\int -2t^{3}\mathrm{d}t$$

Factor out the constant in each of the terms.

$$45\int t^{2}\mathrm{d}t-2\int t^{3}\mathrm{d}t$$

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 45 times \frac{t^{3}}{3}.

$$15t^{3}-2\int t^{3}\mathrm{d}t$$

Since \int t^{k}\mathrm{d}t=\frac{t^{k+1}}{k+1} for k\neq -1, replace \int t^{3}\mathrm{d}t with \frac{t^{4}}{4}. Multiply -2 times \frac{t^{4}}{4}.

$$15t^{3}-\frac{t^{4}}{2}$$

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.

$$15\times 2^{3}-\frac{2^{4}}{2}-\left(15\times 1^{3}-\frac{1^{4}}{2}\right)$$

Simplify.

$$\frac{195}{2}$$