# The volume of solid with vertices located at (1,0), (0,1), (-1,0), (0,-1).

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 6, Problem 22RE

(a)

To determine

## To Find: The volume of solid with vertices located at (1,0), (0,1), (-1,0), (0,-1).

Expert Solution

The volume of the solid is π3_.

### Explanation of Solution

Given information:

The base of solid is square with vertices located at (1,0), (0,1), (-1,0), (0,-1).

Calculation:

Consider that the solid to the right of the origin y=x+1.

Radius of the semicircle is r=x+1.

Find the area of the semicircle using the formula.

A=12πr2.

Expression to find the volume of solid as shown below.

Due to symmetry integrate from 0 to 1 as shown below.

V=2abA(x)dx (1)

Substitute 0 for a, 1 for b, and 12π(1x2) for A(x) in Equation (1).

V=20112π(1x)2dx=0112π(1x)2dx=[π(1x)33]01=π3[((1x)3)]01

=π3[((11)3)]=π3

Therefore, the volume of the solid is π3_.

(b)

To determine

### To find: the volume of the solid.

Expert Solution

the volume of the solid is π3_.

### Explanation of Solution

The solid is modified as a cone by cutting method.

Procedure to obtain the volume of solid is explained below.

1. 1. Cut the solid that passes through to the y axis, and perpendicular to the x axis.
2. 2. Fold half of the solid that lies in the region x0 , so that the point (1,0) touched the point (1,0).
3. 3. The final shape of the solid id right circular cone.
4. 4. The radius of the right circular cone is 1, and the vertex (x,y,x)=(1,0,0).

Calculation:

Find the volume of the cone as shown below.

V=13πr2h (2)

Substitute 1 for r and 1 for h in Equation (2).

V=13π(1)(1)=π3

Therefore, the volume of the cone is π3_.

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