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

Answer 1

#32/21#

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

the volume is simply

#int int \ z(x,y) \ dA = int int \( 4 - 7x - y) \ dA#

it can be done as

#int_{y = 0}^{4} \ int_{x =0 }^{(4-y)/7} \ dx \ dy qquad ( 4 - 7x - y )#

#int_{x = 0}^{4/7} \ int_{y =0 }^{4-7x} \ dy \ dx qquad ( 4 - 7x - y )#

in each case comes out at #32/21#

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Answer 2

Ezekiel Alexander

To find the volume of the solid bounded by the coordinate planes and the plane (7x+y+z=4), you can set up a triple integral over the region enclosed by the planes. Since the solid is bounded by the coordinate planes, its volume can be determined by integrating the constant function 1 over this region. The limits of integration for (x), (y), and (z) will depend on the intersection points of the given plane with the coordinate planes. You would integrate 1 with respect to (x), (y), and (z) over these limits to find the volume.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.