Solve ∫ (from 0 to 8) of (-1.3x)(-0.08x)/2 wrt x

$$\int_{ 0 }^{ 8 } \frac{ (-1.3x)(-0.08x) }{ 2 } d x$$

Evaluate

$$\frac{3328}{375}\approx 8.874666667$$

Solution Steps

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

$$\int _{0}^{8}\frac{-1.3x^{2}\left(-0.08\right)}{2}\mathrm{d}x$$

Multiply -1.3 and -0.08 to get 0.104.

$$\int _{0}^{8}\frac{0.104x^{2}}{2}\mathrm{d}x$$

Divide 0.104x^{2} by 2 to get 0.052x^{2}.

$$\int _{0}^{8}0.052x^{2}\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int \frac{13x^{2}}{250}\mathrm{d}x$$

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

$$\frac{13\int x^{2}\mathrm{d}x}{250}$$

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}.

$$\frac{13x^{3}}{750}$$

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.

$$\frac{13}{750}\times 8^{3}-\frac{13}{750}\times 0^{3}$$

Simplify.

$$\frac{3328}{375}$$