Solve ∫ (from 0 to 4) of (6.67x)(-0.08x) wrt x

$$\int_{ 0 }^{ 4 } (6.67x)(-0.08x) d x$$

Evaluate

$$-\frac{21344}{1875}\approx -11.383466667$$

Solution Steps

Multiply x and x to get x^{2}.

$$\int _{0}^{4}6.67x^{2}\left(-0.08\right)\mathrm{d}x$$

Multiply 6.67 and -0.08 to get -0.5336.

$$\int _{0}^{4}-0.5336x^{2}\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int -\frac{667x^{2}}{1250}\mathrm{d}x$$

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

$$-\frac{667\int x^{2}\mathrm{d}x}{1250}$$

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}.

$$-\frac{667x^{3}}{3750}$$

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.

$$-\frac{667}{3750}\times 4^{3}+\frac{667}{3750}\times 0^{3}$$

Simplify.

$$-\frac{21344}{1875}$$