Solve ∫ (from 0 to 4) of (x^2-6.6x)(-0.1x) wrt x

$$\int_{ 0 }^{ 4 } ( { x }^{ 2 } -6.6x)(-0.1x) d x$$

Evaluate

$$7.68$$

Solution Steps

Use the distributive property to multiply x^{2}-6.6x by -0.1.

$$\int _{0}^{4}\left(-0.1x^{2}+0.66x\right)x\mathrm{d}x$$

Use the distributive property to multiply -0.1x^{2}+0.66x by x.

$$\int _{0}^{4}-0.1x^{3}+0.66x^{2}\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int -\frac{x^{3}}{10}+\frac{33x^{2}}{50}\mathrm{d}x$$

Integrate the sum term by term.

$$\int -\frac{x^{3}}{10}\mathrm{d}x+\int \frac{33x^{2}}{50}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$-\frac{\int x^{3}\mathrm{d}x}{10}+\frac{33\int x^{2}\mathrm{d}x}{50}$$

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 -0.1 times \frac{x^{4}}{4}.

$$-\frac{x^{4}}{40}+\frac{33\int x^{2}\mathrm{d}x}{50}$$

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 0.66 times \frac{x^{3}}{3}.

$$-\frac{x^{4}}{40}+\frac{11x^{3}}{50}$$

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.

$$-\frac{4^{4}}{40}+\frac{11}{50}\times 4^{3}-\left(-\frac{0^{4}}{40}+\frac{11}{50}\times 0^{3}\right)$$

Simplify.

$$\frac{192}{25}$$