Question

The base of a solid in the first quadrant of the xy plane is a right triangle bounded by the coordinate axes and the line x + y = 2. cross sections of the solid perpendicular to the base are squares. what is the volume, in cubic units, of the solid?

Answer 1

The area of an equilateral triangle of side "s" is s^2*sqrt(3)/4. So the volume of the slices in your problem is

(x - x^2)^2 * sqrt(3)/4.

Integrating from x = 0 to x = 1, we have

[(1/3)x^3 - (1/2)x^4 + (1/5)x^5]*sqrt(3)/4

= (1/30)*sqrt(3)/4 = sqrt(3)/120 = about 0.0144.

Since this seems quite small, it makes sense to ask what the base area might be...integral from 0 to 1 of (x - x^2) dx = (1/2) - (1/3) = 1/6. Yes, OK, the max height of the triangles occurs where x - x^2 = 1/4, and most of the triangles are quite a bit shorter...

Answer 2

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

Step-by-step explanation:

Visualization:

Imagine a right triangle with one leg on the x-axis and the other on the y-axis. The hypotenuse of the triangle intersects the line x + y = 2 at a point (x, y). The solid is formed by stacking square cross-sections perpendicular to the base triangle, with each square having a side length equal to the distance between the line x + y = 2 and the triangle's hypotenuse at that point.

Volume Calculation:

Let x be the distance from the origin to the point where the hypotenuse intersects the line x + y = 2. Then, the length of the side of each square cross-section is (2 - x). The area of each cross-section is therefore (2 - x)^2.

As we move from the origin to the vertex of the triangle where the hypotenuse intersects the line, the distance x increases from 0 to 2. Thus, the volume of the solid can be calculated by integrating the area of the cross-sections over the range of x:

Volume = ∫(2 - x)^2 dx from x = 0 to x = 2

Integration:

Solving the integral using the power rule, we get:

Volume = [4/3 * x^3 - 2x^2 + x] from x = 0 to x = 2

Evaluating the expression at the limits of integration:

Volume = (32/3 - 8 + 2) - (0 - 0 + 0) = 32/3 - 6 = 8/3 cubic units

Therefore, the volume of the solid is 8/3 cubic units.