Consider the solid S described below. The base of S is the triangular region with vertices (0, 0), (2, 0), and (0, 2). Cross-sections perpendicular to the x-axis are squares. Find the volume V of this solid.

To answer that question, we need to get the equation of the hypothenuse (L) of the right triangle.

The solution is:

V = 4/3 cubic units

As the cross-sections perpendicular (pieces) to the x-axis are squares, the volume (V) of the solid,  is the product of the area of the base ( the square ) times dx. We will integrate from 0 to the point ( 2 , 0 ).

Then

The straight line passing through P ( 2, 0 ) and Q ( 0 , 2) is:

y = m×x + b

we need to find m and b

m = ( 2 - 0 ) / ( 0 - 2 )     m = - 1

P  ( 2, 0 )      ⇒    y = m×x + b  ⇒   0 = -1× (2) + b       ⇒  b = 2

The equation of the hypothenuse is:

y = - x  + 2           ⇒   x = 2 - y

And  this is the side of each square, then

dV = A × dx    ⇒  ∫dV = ∫ A×dx

V = ∫ A×dx    

A is the area of the square of side:  y = 2 - x; then

V = ∫ ₀² ( 2 - x )² × dx

V = ∫ ₀² ( 4 + x² -4×x ) × dx

V = 4×x + (1/3)×x³ - 2×x² |₀²      V = 4×2 + (1/3)×(2)² - 4×(2) - 0

V = 4/3 cubic units

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is the region in the plane bounded by lines , , and or .

Each cross section, for a given value of in the interval [0, 2], has an area equal to , where is the length of the square section's base.

Then the volume of the solid is

hi,

you half the rectangle that you ae asking about so then you would see the scale factor of 1/2 the one u have selected, the one on top is half.

hope this !