Question

Find the volume of the given solid. Bounded by the planes z=x,y=x,x+y=4 and z=0

Answer 1

To find the volume of the given solid bounded by the planes, set up a triple integral with appropriate limits of integration for x, y, and z. Evaluate the integral to find the volume.

Step-by-step explanation:

To find the volume of the given solid bounded by the planes, we need to determine the limits of integration for each variable and set up the triple integral.

First, let's find the limits of integration for x, y, and z:

• For x: from 0 to 4-y
• For y: from 0 to 4-x
• For z: from 0 to x

Then, we can set up the triple integral:

V = \int_0^4 \int_0^{4-x} \int_0^x dzdydx

You can evaluate this integral to find the volume of the solid.

Answer 2

The volume of the given solid is 8/3 cubic units.

How the volume of the solid was calculated

To find the volume of the solid bounded by the planes z = x, y = x, (x + y = 4), and z = 0

Let's set up a triple integral using the given bounds.

The region in the xy-plane is a triangle formed by the lines (y = x), (x + y = 4), and the x-axis.

y = x intersects x + y = 4

when x = 2

The limits for x is from 0 to 2.

For y it goes from 0 to x within this region.

Now, for z, it goes from z = 0 to z = x.

The volume V can be expressed as the triple integral:

V = ∫₀²∫₀ˣ∫₀ˣ{dzdydx}

Evaluate this integral to find the volume. Starting with the innermost integral:

∫₀ˣdz = x

Then, integrate with respect to y.

∫₀ˣ xdy = xy

[xy]₀ˣ = x²

Integrate with respect to x.

∫₀²x²dx = [1/3x³]₀²

= 1/3(2³) - 1/3(0³)

= 8/3 - 0

= 8/3

So, the volume of the given solid is 8/3 cubic units