Solve ∫ (from 0 to 4) of (-1.35x^2+0.45x^3) wrt x

$$\int_{ 0 }^{ 4 } (-1.35 { x }^{ 2 } +0.45 { x }^{ 3 } ) d x$$

Evaluate

$$0$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \frac{-27x^{2}+9x^{3}}{20}\mathrm{d}x$$

Integrate the sum term by term.

$$\int -\frac{27x^{2}}{20}\mathrm{d}x+\int \frac{9x^{3}}{20}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\frac{-27\int x^{2}\mathrm{d}x+9\int x^{3}\mathrm{d}x}{20}$$

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 -1.35 times \frac{x^{3}}{3}.

$$\frac{-9x^{3}+9\int x^{3}\mathrm{d}x}{20}$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{3}\mathrm{d}x with \frac{x^{4}}{4}. Multiply 0.45 times \frac{x^{4}}{4}.

$$-\frac{9x^{3}}{20}+\frac{9x^{4}}{80}$$

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.

$$-\frac{9}{20}\times 4^{3}+\frac{9}{80}\times 4^{4}-\left(-\frac{9}{20}\times 0^{3}+\frac{9}{80}\times 0^{4}\right)$$

Simplify.

$$0$$