Solve ∫ (from - 1 to 1) of int_{x^2}^1int_0^1-y1 wrt zdydx

$$\int _ { - 1 } ^ { 1 } \int _ { x ^ { 2 } } ^ { 1 } \int _ { 0 } ^ { 1 - y } 1 d z d y d x$$

Evaluate

$$\frac{8}{15}\approx 0.533333333$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \int _{x^{2}}^{1}\int _{0}^{1-y}1\mathrm{d}z\mathrm{d}y\mathrm{d}x$$

Simplify.

$$\frac{x}{2}-\frac{x^{3}}{3}+\frac{x^{5}}{10}$$

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.

$$\frac{1}{2}\times 1-\frac{1^{3}}{3}+\frac{1^{5}}{10}-\left(\frac{1}{2}\left(-1\right)-\frac{\left(-1\right)^{3}}{3}+\frac{\left(-1\right)^{5}}{10}\right)$$

Simplify.

$$\frac{8}{15}$$