Step by step solution to "Determine the volume of the sold in the..."

Rui Baldwin

Answered question

2020-10-20

Determine the volume of the sold in the 1st octant bounded by

x + y + 2 z = 4

Anonym

Skilled2020-10-21Added 108 answers

Step 1
The given equation is:

x + y + 2 z = 4

The above equation is solved as:

2 z = 4 - x - y

z = 2 - \frac{x}{2} - \frac{y}{2}

T h u s, z v a r i e s \in 0 \leq z \leq 2 - \frac{x}{2} - \frac{y}{2}

In XY plane , the equation is:
x+y=4
Thus, y varies in

0 \leq y \leq 4 - x

and x varies in

0 \leq x \leq 4

Step 2
The volume is given as:

V o l u m e = \int_{0}^{4} \int_{0}^{4 - x} \int_{0}^{2 - \frac{x}{2} - \frac{y}{2}} d z d y d x

= \int_{0}^{4} \int_{0}^{4 - x} {[z]}_{0}^{2 - \frac{x}{2} - \frac{y}{2}} d y d x

= \int_{0}^{4} \int_{0}^{4 - x} (2 - \frac{x}{2} - \frac{y}{2}) d y d x

= \int_{0}^{4} {[2 y - \frac{x y}{2} - \frac{y^{2}}{4}]}_{0}^{4 - x} d x

= \int_{0}^{4} [2 (4 - x) - \frac{x (4 - x)}{2} - \frac{{(4 - x)}^{2}}{4}] d x

= \int_{0}^{4} [8 - 2 x - 2 x + \frac{x^{2}}{2} - \frac{(16 - 8 x + x^{2})}{4}] d x

= \int_{0}^{4} [8 - 2 x - 2 x + \frac{x^{2}}{2} - 4 + 2 x - \frac{x^{2}}{4}] d x

= \int_{0}^{4} [4 - 2 x + \frac{x^{2}}{4}] d x

= {[4 x - \frac{2 x^{2}}{2} + \frac{x^{3}}{12}]}_{0}^{4}

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