Question

Consider the solid S described below. The base of S is the triangular region with vertices (0, 0), (4, 0), and (0, 4). Cross-sections perpendicular to the x-axis are squares. Find the volume V of this solid.

Answer 1

The volume of solid S, with a triangular base and perpendicular square cross-sections, is found by integrating the area of cross-sections from one end of the base to the other. The volume V is 64/3 cubic units.

Step-by-step explanation:

Volume of Solid S with Triangular Base and Square Cross-Sections

To find the volume V of the solid S, we consider that the base is a right triangle in the xy-plane with vertices at (0, 0), (4, 0), and (0, 4). The cross-sections perpendicular to the x-axis are squares, meaning that their side lengths vary linearly with x along the base of the triangle.

To calculate the volume, we integrate the area of the square cross-sections from the left vertex at x=0 to the right vertex at x=4. The side length of each square is given by the y-coordinate of the triangle at that x-value, which decreases linearly from 4 to 0 as x increases from 0 to 4. This relationship can be expressed as l(x) = 4 - x. The area A of a square is A = s² where s is the side length, so in this case, A(x) = (4 - x)².

The volume is then the integral of A(x) with respect to x from 0 to 4:

1. Set up the integral for the volume V: ∫ A(x) dx from 0 to 4.
2. Integrate the function A(x) = (4 - x)² to find the area under the curve from x=0 to x=4.
3. Calculate the definite integral, which provides the volume of solid S.

Performing the integration, we find:

∫ A(x) dx = ∫ (4 - x)² dx

= ∫ (16 - 8x + x²) dx from 0 to 4

= [16x - 4x² + x³/3] from 0 to 4

= (16*4 - 4*4² + 4³/3) - (0 - 0 + 0)

= 64/3

After calculating, we find that the volume V is 21.333... or 64/3 cubic units.

Answer 2

Volume =
`(64)/(3)`

Step-by-step explanation:

Given - Consider the solid S described below. The base of S is the triangular region with vertices (0, 0), (4, 0), and (0, 4). Cross-sections perpendicular to the x-axis are squares.

To find - Find the volume V of this solid.

Solution -

Given that,

The equation of the line with both x-intercept and y-intercept as 4 is -

`(x)/(4) + (y)/(4) = 1`

⇒x + y = 4

⇒y = 4 - x

Now,

Volume =
`\int\limits^a_b {A(x)} \, dx`

where

A(x) is the area of general cross-section.

It is given that,

Cross-sections perpendicular to the x-axis are squares.

So,

A(x) = (4 - x)²

As solid lies between x = 0 and x = 4

So,

The Volume becomes

Volume =
`\int\limits^4_0 {(4 - x)^(2) } \, dx`

=
`\int\limits^4_0 {[(4)^(2) + (x)^(2) - 8x] } \, dx`

=
`\int\limits^4_0 {[16 + x^(2) - 8x] } \, dx`

=
`{[16 x + (x^(3))/(3) - (8x^(2) )/(2) ] } ^4_0`

=
`{[16(4 - 0) + (4^(3))/(3) - (0^(3))/(3) - 4 [4^(2) - 0^(2)] ] }`

=
`{[16(4) + (64)/(3) - 0 - 4 [16 - 0] ] }`

=
`{[64 + (64)/(3) - 64 ] }`

=
`(64)/(3)`

⇒Volume =
`(64)/(3)`