Solve ∫ (from 0 to 3) of x^2-2pi

$$\int _ { 0 } ^ { 3 } x ^ { 2 } - 2 \pi$$

Evaluate

$$9-6\pi \approx -9.849555922$$

Solution Steps

Evaluate the indefinite integral first.

$$\int x^{2}-2\pi \mathrm{d}x$$

Integrate the sum term by term.

$$\int x^{2}\mathrm{d}x+\int -2\pi \mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int x^{2}\mathrm{d}x-2\int \pi \mathrm{d}x$$

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}.

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

Find the integral of \pi using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{x^{3}}{3}-2\pi x$$

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.

$$\frac{3^{3}}{3}-2\pi \times 3-\left(\frac{0^{3}}{3}-2\pi \times 0\right)$$

Simplify.

$$9-6\pi$$