Solve ∫ (from 0 to 5) of int_0^3-xint_{0}^4-y^24 wrt zdydx

$$\int _ { 0 } ^ { 5 } \int _ { 0 } ^ { 3 - x } \int _ { 0 } ^ { 4 - y ^ { 2 } } 4 d z d y d x$$

Evaluate

$$\frac{55}{3}\approx 18.333333333$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \int _{0}^{3-x}\int _{0}^{4-y^{2}}4\mathrm{d}z\mathrm{d}y\mathrm{d}x$$

Simplify.

$$12x+10x^{2}-4x^{3}+\frac{x^{4}}{3}$$

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.

$$12\times 5+10\times 5^{2}-4\times 5^{3}+\frac{5^{4}}{3}-\left(12\times 0+10\times 0^{2}-4\times 0^{3}+\frac{0^{4}}{3}\right)$$

Simplify.

$$\frac{55}{3}$$