Solve ∫ (from 7 to 10) of (0.012x^2-0.0012x^3) wrt x

$$\int_{ 7 }^{ 10 } (0.012 { x }^{ 2 } -0.0012 { x }^{ 3 } ) d x$$

Evaluate

$$0.3483$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \frac{3x^{2}}{250}-\frac{3x^{3}}{2500}\mathrm{d}x$$

Integrate the sum term by term.

$$\int \frac{3x^{2}}{250}\mathrm{d}x+\int -\frac{3x^{3}}{2500}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\frac{3\int x^{2}\mathrm{d}x}{250}-\frac{3\int x^{3}\mathrm{d}x}{2500}$$

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

$$\frac{x^{3}}{250}-\frac{3\int x^{3}\mathrm{d}x}{2500}$$

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 -0.0012 times \frac{x^{4}}{4}.

$$\frac{x^{3}}{250}-\frac{3x^{4}}{10000}$$

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.

$$\frac{10^{3}}{250}-\frac{3}{10000}\times 10^{4}-\left(\frac{7^{3}}{250}-\frac{3}{10000}\times 7^{4}\right)$$

Simplify.

$$\frac{3483}{10000}$$