Q. 27

Question

Think about the area between the x-axis on $[0, 4]$ and $f (x) = \sqrt{x}$ . Use four shells based on the provided rectangles to approximate the volume of the resulting solid for each line of rotation given in exercise.

around the $y = - 1$ line

Step-by-Step Solution

Verified

Answer

The volume of the solid formed is $59.691$

1Step 1: Given information

The line $y = - 1$

2Step 2: Calculation

The rectangles are drawn parallel to the x-axis, forming shells with bases on the y-axis, as the solid is rotated around the line $y = - 1$ .

The range $[0, 2]$ on the y-axis is divided into four subintervals: $[0, 0.5], [0.5, 1], [1, 1.5],$ and $[1.5, 2] .$

Each subinterval has a width of $0.5$ , or $Δ y = 0.5$ .

Since $f (x) = \sqrt{x}$ is the curve and the region in the diagram is constrained to $[0, 4]$ ,

the height of the shells is given by $h = 4 - f^{- 1} (x) = 4 - y^{2}$

The first shell on the interval $[0, 0.5]$ has a radius of $0.25 - (- 1) = 1.25$ and a height of $h = 4 - (0.25)^{2} = 3.9375$ , which is found near the middle of the interval.

Thus, the first shell's volume equals

$2 π r h Δ y = 2 π (1.25) (3.9375) (0.5) = 15.463$

Volume of first shell is $15.463$

The average radius of the second shell is $0.75 - (- 1) = 1.75$ and its height at the midpoint is determined as $h = 4 - (0.75)^{2}$ , which is $h = 3.4375 .$

Consequently, the second shell's volume is

$2 π r h Δ y = 2 π (1.75) (3.4375) (0.5) = 18.899$

Volume of second shell is $18.899$

The average radius of the third shell is $1.25 - (- 1) = 2.25$ and its height at the midpoint is determined as $h = 4 - (1.25)^{2},$ that is $h = 2.4375$

Consequently, the third shell's volume is

$2 π r h Δ y = 2 π (2.25) (2.4375) (0.5) = 17.2297$

The average radius of the fourth shell is $1.75 - (- 1) = 2.75$ and its height at the midpoint is determined as $h = 4 - (1.75)^{2}$ , which is $h = 0.9375 .$