Solve ∫ (from - 70 to 70) of 116-0.1x^2

$$\int _ { - 70 } ^ { 70 } 116 - 0.1 x ^ { 2 }$$

Evaluate

$$-\frac{19880}{3}\approx -6626.666666667$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 116-\frac{x^{2}}{10}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 116\mathrm{d}x+\int -\frac{x^{2}}{10}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int 116\mathrm{d}x-\frac{\int x^{2}\mathrm{d}x}{10}$$

Find the integral of 116 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$116x-\frac{\int x^{2}\mathrm{d}x}{10}$$

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 -0.1 times \frac{x^{3}}{3}.

$$116x-\frac{x^{3}}{30}$$

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.

$$116\times 70-\frac{70^{3}}{30}-\left(116\left(-70\right)-\frac{\left(-70\right)^{3}}{30}\right)$$

Simplify.

$$-\frac{19880}{3}$$