Solve ∫ (from 0 to 3) of (9-x^2) wrt x

$$\int _ { 0 } ^ { 3 } ( 9 - x ^ { 2 } ) d x$$

Evaluate

$$18$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 9-x^{2}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 9\mathrm{d}x+\int -x^{2}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int 9\mathrm{d}x-\int x^{2}\mathrm{d}x$$

Find the integral of 9 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$9x-\int x^{2}\mathrm{d}x$$

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}. Multiply -1 times \frac{x^{3}}{3}.

$$9x-\frac{x^{3}}{3}$$

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.

$$9\times 3-\frac{3^{3}}{3}-\left(9\times 0-\frac{0^{3}}{3}\right)$$

Simplify.

$$18$$