# How do you find the volume of the solid bounded by the coordinate planes and the plane 7x+y+z=4?

Jul 14, 2016

$\frac{32}{21}$

#### Explanation:

the drawing is key. start by finding the intercepts with each of the axes, the intercept line on the xy plane follows as $y + 7 x = 4$

the volume is simply

$\int \int \setminus z \left(x , y\right) \setminus \mathrm{dA} = \int \int \setminus \left(4 - 7 x - y\right) \setminus \mathrm{dA}$

it can be done as

${\int}_{y = 0}^{4} \setminus {\int}_{x = 0}^{\frac{4 - y}{7}} \setminus \mathrm{dx} \setminus \mathrm{dy} q \quad \left(4 - 7 x - y\right)$

OR

${\int}_{x = 0}^{\frac{4}{7}} \setminus {\int}_{y = 0}^{4 - 7 x} \setminus \mathrm{dy} \setminus \mathrm{dx} q \quad \left(4 - 7 x - y\right)$

in each case comes out at $\frac{32}{21}$