Question

To find: The volume of the resulting solid by the region bounded by given curve by the use of the method of shell.

Short answer

The volume of the resulting solid by the region bounded by given curve is \(134.041\).

Step-by-step solution

Given information

The equation is \(x = {(y - 3)^2},x = 4;\) about \(y = 1\).

The concept of method of washer

The shell method calculates the volume of the full solid of revolution by summing the volumes of these thin cylindrical shells as the thickness\(\Delta x\backslash \)Delta\(x\Delta x\)goes to 000 in the limit:\(V = \int d V = \int a b2\pi xydx = \int a b2\pi xf(x)dx\).

Draw the graph

Calculate the height

\(x = {(y - 3)^2},\quad x = 4;\quad \) about \(y = 1\)

The cylinder method will probably be easier. An example cylinder is shown in red.

\(\begin{aligned}{}4 = {(y - 3)^2}\\2 = \pm (y - 3)\\y = 3 \pm 2 = 1,5\end{aligned}\)

By the use of the cylinder method, we will integrate along the \(y\)-axis from 1 to 5.

The radius of each cylinder will be \(r = y - 1\) and the height will be

\(h = 4 - {(y - 3)^2} = 4 - \left( {{y^2} - 6y + 9} \right) = - {y^2} + 6y - 5\).

Calculate the volume

\(\begin{aligned}{}V = \int_a^b 2 \pi rhdy = 2\pi \int_1^5 {(y - 1)} \left( { - {y^2} + 6y - 5} \right)dy\\ = 2\pi \int_1^5 {\left( { - {y^3} + 6{y^2} - 5y + {y^2} - 6y + 5} \right)} dy\\ = 2\pi \int_1^5 {\left( { - {y^3} + 7{y^2} - 11y + 5} \right)} dy\\ = 2\pi \left( { - \frac{1}{4}{y^4} + 7 \cdot \frac{1}{3}{y^3} - 11 \cdot \frac{1}{2}{y^2} + 5y} \right)_1^5\end{aligned}\)

Simplify further,

\(\begin{aligned}{}V = 2\pi \left( { - \frac{1}{4}{{(5)}^4} + \frac{7}{3} \cdot {{(5)}^3} - \frac{{11}}{2} \cdot {{(5)}^2} + 25 - \left( { - \frac{1}{4} + \frac{7}{3} - \frac{{11}}{2} + 5} \right)} \right)\\ = 2\pi \left( { - \frac{{625}}{4} + \frac{{875}}{3} - \frac{{275}}{2} + 25 + \frac{1}{4} - \frac{7}{3} + \frac{{11}}{2} - 5} \right)\\ = 2\pi \left( { - \frac{{624}}{4} + \frac{{868}}{3} - \frac{{264}}{2} + 20} \right)\\ = 2\pi \left( {\frac{{ - 3(624) + 4(868) - 6(264) + 12(20)}}{{12}}} \right)\end{aligned}\)

Simplify further,

\(\begin{aligned}{}V = 2\pi \left( {\frac{{256}}{{12}}} \right)\\ = \frac{{256\pi }}{6}\\ = \frac{{128\pi }}{3}\\ \approx 134.041\end{aligned}\)

The volume of the resulting solid by the region bounded by given curve is \(134.041\).