Mathematics: Somebody answer this because I m giving out points and don t want it to get deleted

Somebody answer this because I m giving out points - id14505461, mauricestepenson791

Marleneg

12.07.2020

Volume = $\frac{20}{3}$ cubic units

Step-by-step explanation: Given the equation find the limits:

For z:

8x + y + z = 4

z = 4 - 8x - y

limits 0 < z < 4-8x-y

For y:

4 - 8x - y = 0

y = 4 - 8x

limits: 0 < y < 4-8x

For x:

4 - 8x = 0

x = 1/2

limits: 0 < x < 1/2

Calculating triple integral and determining volume:

V = $\int\ {\int\ {\int\ {} \, dz } \, dy } \, dx$

V = $\int\ {\int\ {z} \, dy } \, dx$

V = $\int\ {\int\ (4-8x-y)-0 \, dy } \, dx$

V = $\int\ { (4y-8xy-\frac{y^{2}}{2} ) } \, dx$

V = $\int\limits^a_0 {4(4-8x)-8x(4-8x)-\frac{(4-8x)^{2}}{2} - 0 } \, dx$

V = $\int\limits^a_0 {8+32x-32x^{2} } \, dx$

V = $8x+16x^{2}-\frac{32}{3}x^{3}$

V = $8(\frac{1}{2} )+16(\frac{1}{2})^{2} -\frac{32}{3}(\frac{1}{2}) ^{3}$

V = $\frac{20}{3}$

Volume of the solid is V = $\frac{20}{3}$ cubic units.

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