#### To determine

(a)

**To find:**

The volume of a solid whose base is a square with vertices located at 1, 0, 0, 1, -1, 0, and (0, -1), and cross section perpendicular to the x-axis is a semicircle.

#### Answer

π3

#### Explanation

**1) Concept:**

Use the definition of volume.

**2) 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

**3) Given:**

Vertices of square are 1, 0, 0, 1, -1, 0 and (0, -1)

**4) Calculations:**

From the figure, one cross section perpendicular to x-axis represents the diameter of a semicircle.

The equation of line representing the upper left part is

-x+y=1

Solving for y,

y=x+1

From the figure, the radius of the semicircle is r=x+1, -1≤x≤0.

Area of semicircle is

A=12πr2=12πx+12

Use symmetry to find volume of a solid S, integrate area of semicircles from 0 to 1.

V=2∫-1012πx+12 dx

After expanding the square term,

V=π∫-101+2x+x2dx

After integrating,

V=πx+x2+x330-1

Applying the Fundamental Theorem of Calculus,

V=π-1+1+13=π3

**Conclusion:**

The volume of a solid whose base is a square with vertices located at 1, 0, 0, 1, -1, 0, and (0, -1) and cross section perpendicular to the x-axis is a semicircle is π3.

#### To determine

(b)

**To show:**

The volume of the solid of part (a) can be computed more simply by first cutting the solid and rearranging it to form a cone.

#### Answer

The volume of the solid of part (a) can be computed more simply by first cutting the solid andrearranging it to form a cone.

#### Explanation

By cutting the solid such that it is passing through y-axis and perpendicular to plane x-axis such that half fold is in the region x≤0. The point -1, 0 touches the point (1, 0).

After this rearrangement, the solid will be a cone with the radius and height 1.

By the formula for volume of a cone,

V=13πr2h

Since r=1, h=1,

V=13π·12·1=π3

**Conclusion:**

The volume of the solid of part (a) can be computed more simply by first cutting the solid andrearranging it to form a cone.