Question

The region bounded by the curves y=(1/x), y=x^2, and x = 0.1 is revolved around the y-axis to form a solid. What is the approximate volume of the solid?

Answer 1

To find the volume of the solid formed by revolving the region bounded by the curves y=1/x, y=x^2, and x=0.1 around the y-axis, we can use the method of cylindrical shells.

Step-by-step explanation:

To find the volume of the solid formed by revolving the region bounded by the curves y=1/x, y=x^2, and x=0.1 around the y-axis, we can use the method of cylindrical shells.

We will divide the region into infinitesimally thin cylindrical shells of height ∆x and radius y. The volume of each shell is given by 2πxy∆x.

Integrating the volume of the shells from x=0.1 to x=1 will give us the total volume of the solid. So, the approximate volume of the solid is given by the integral of 2πxy∆x from 0.1 to 1.

Answer 2

C

Step-by-step explanation:

1/x = x²

x³ = 1

x = 1

y = 1

x = 0.1

1/0.1 = 10

0.1² = 0.01

V1: x² = y

Integrate: y²/2

Limits y = 0.01 to y = 1

½(1² - 0.01²) = 0.49995 pi

V2: x² = 1/y² = y^-2

Integrate: -1/y

Limits y = 1 to y = 10

(-1/10) - (-1/1)

0.9pi

V1 + V2 - [pi × 0.1² × (10-0.01)]

0.49995pi + 0.9pi - 0.0999pi

= 1.30005 pi units³

= 4.084227529 units³