Solve ∫ (from - 2 to 1) of (1-t^2)dt

$$\int _ { - 2 } ^ { 1 } ( 1 - t ^ { 2 } ) d t$$

Evaluate

$$0$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 1-t^{2}\mathrm{d}t$$

Integrate the sum term by term.

$$\int 1\mathrm{d}t+\int -t^{2}\mathrm{d}t$$

Factor out the constant in each of the terms.

$$\int 1\mathrm{d}t-\int t^{2}\mathrm{d}t$$

Find the integral of 1 using the table of common integrals rule \int a\mathrm{d}t=at.

$$t-\int t^{2}\mathrm{d}t$$

Since \int t^{k}\mathrm{d}t=\frac{t^{k+1}}{k+1} for k\neq -1, replace \int t^{2}\mathrm{d}t with \frac{t^{3}}{3}. Multiply -1 times \frac{t^{3}}{3}.

$$t-\frac{t^{3}}{3}$$

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.

$$1-\frac{1^{3}}{3}-\left(-2-\frac{\left(-2\right)^{3}}{3}\right)$$

Simplify.

$$0$$