#### To determine

**To find:**

The volume of a solid S whose base is the triangular region with vertices 0, 0, 1, 0 and 0, 1 and cross-sections perpendicular to the *x*-axis are squares.

#### Answer

13

#### Explanation

**1) Concept:**

Definition of volume:

Let S be a solid that lies between x=a and x=b. If the cross sectional area of S in the plane Px, through x and perpendicular to the x-axis, is A(x), where A is continuous function, then the volume of S is

V=limn→∞∑i=1nAxi*∆x=∫abAxdx

**2) Given:**

Vertices of the base triangle are 0, 0, 1, 0 and 0, 1.

**3) Calculations:**

A cross section represents the side s(red line) of the square.

The base of the cross section corresponding to the point x has length 1-x. Because cross sections are perpendicular to the x-axis and they are given by the distance between lines y=0 and y=x-1.

The area of cross section that is the area of square is Ax=side2=x-12

Cross section are perpendicular to x-axis, therefore, we shall integrate cross sections with respect to x-axis. The boundary points are 0 and 1. Thus

V=∫01A dy=∫011-x2dx

Expand the term 1-x2,

V=∫01(1-2x+x2)dx

After integrating,

V=x-x2+x3310

Applying the Fundamental Theorem of Calculus

V=1-12+133-0-0+0=13

**Conclusion:**

The volume of a solid S whose base is the triangular region with vertices 0, 0, 1, 0 and 0, 1 and cross-sections perpendicular to the *x*-axis are squares is 13