Learn volume using integration of general solids, washer method and cylindrical shells method. Practice washer & shell method calculus formula.

- Let S be the solid that lies between \(x=a\) and \(x=b\). If the cross sectional area of \(S\) in plane \(P_x\) through \(x\) and \(\perp\) to x-axis is \(A(x)\) where \(A\) is a continuous function, then volume of \(S\)is given by \(V=\int\limits_a^b\,A(x)\;dx\)

- We cut S into pieces and obtain a plane require that is called cross section of S. This cross sectional Area \(A(x)\) will vary as \(x\) increases function \('a'\) to \('b'\).
- Divide S into \(x\) slabs of equal width \(\Delta x\) by using planes \(P_{x_1}, P_{x_2},....,P_{x_n}\) to slice the solid. The \(i^{th}\) slab is like a cylinder with base
- Area = \(A(x_i^*)\,\big(x_i^*\in(x_{i-1}, \;x_i)\big)\) and height \(\Delta x\).

\(\therefore \) Volume of \(i^x\) slab = \(V(S_i)\approx A(x_i^*)\,\Delta x\)

Adding the volume

\(V\approx\sum\limits_{i=1}^{n}\,A(x_i^*)\,\Delta x\)

as \(n\to\infty\)

\(V=\int\limits_a^b\,A(x)\,dx\)

A \(\pi\left ( \dfrac {4\sqrt 2-3}{2} \right)\)

B \(\pi\left ( \dfrac {\sqrt 2-1}{3} \right)\)

C \(\dfrac {22\pi}{3}\)

D \((\sqrt 2 + \sqrt 3)\pi\)

- Let S be the solid that lies between \(x=a\) and \(x=b\). If the cross sectional area of \(S\) in plane \(P_x\) through \(x\) and \(\perp\) to x-axis is \(A(x)\) where, \(A\) is a continuous function, then volume of \(S\)is given by \(V=\int\limits_a^b\,A(x)\;dx\)

- We cut S into pieces and obtain a plane require that is called cross section of S. This cross sectional Area \(A(x)\) will vary as \(x\) increases function \('a'\) to \('b'\).
- Divide S into \(x\) slabs of equal width \(\Delta x\) by using planes \(P_{x_1}, P_{x_2},....,P_{x_n}\) to slice the solid. The \(i^{th}\) slab is like a cylinder with base
- Area = \(A(x_i^*)\,\big(x_i^*\in(x_{i-1}, \;x_i)\big)\) and height \(\Delta x\).

\(\therefore \) Volume of \(i^x\) slab = \(V(S_i)\approx A(x_i^*)\,\Delta x\)

Adding the volume

\(V\approx\sum\limits_{i=1}^{n}\,A(x_i^*)\,\Delta x\)

as \(n\to\infty\)

\(V=\int\limits_a^b\,A(x)\,dx\)

A \(\dfrac {128\,\pi}{3}\)

B \(\dfrac {5\,\pi}{2}\)

C \(\dfrac {56\pi}{3}\)

D \(\dfrac {512\,\pi}{7}\)

- Some volumes are difficult to calculate by the method of disk or washers.
- Consider the volume of solid obtained by rotating region bounded by \(y=0\), \(y=3x^2-x^3\) about y-axis.

- Since \(x\)is difficult to evaluate in terms of \(y\) , the method of disks is not very useful.
- Consider a cylindrical shell with inner radius \(r_1\) and outer radius \(r_2\).

Volume of shell = Volume of outer cylinder – Volume of inner cylinder

\(=V_2-V_1\\=\pi\,r_2^2\,h-\pi r_1^2\,h\)

\(=\pi\,h(r_2^2-r_1^2)\)

\(=\pi\,h(r_2+r_1)(r_2-r_1)\)

\(=2\pi\,h \left(\dfrac {r_1+r_2}{2}\right) (r_2-r_1)\)

\(=2\pi\,r\,h \;\Delta r\)

where, \(\Delta r=r_2-r_1\)

\(\therefore\) Volume of cylindrical shell \(=2\pi\,r\,h \;\Delta r\)

- Now consider the solid obtained by rotating about y-axis, the region bounded by \(y=f(x) \Big(f(x) \geq 0 \Big)\) \(y=0\), \(x=a\) and \(x=b\) when \(b>a \geq 0\)

The volume of solid obtained is

\(V=\displaystyle\int_a^b\,2\pi x\,f(x)\;dx\) where \(0\leq a <b\)

Cylindrical shell \(\rightarrow\;2\pi\,r\,h \;\Delta r\)

A \(\dfrac {7\,\pi}{15}\)

B \(\dfrac {\pi}{2}\)

C \(\dfrac {5\,\pi}{3}\)

D \(\dfrac {6\,\pi}{5}\)

- Some volumes are difficult to calculate by the method of disk or washers.
- Consider the volume of solid obtained by rotating region bounded by \(y=0\), \(y=3x^2-x^3\) about y-axis.

- Since \(x\)is difficult to evaluate in terms of \(y\) , the method of disks is not very useful.
- Consider a cylindrical shell with inner radius \(r_1\) and outer radius \(r_2\).

Volume of shell = Volume of outer cylinder – Volume of inner cylinder

\(=V_2-V_1\\=\pi\,r_2^2\,h-\pi r_1^2\,h\)

\(=\pi\,h(r_2^2-r_1^2)\)

\(=\pi\,h(r_2+r_1)(r_2-r_1)\)

\(=2\pi\,h \left(\dfrac {r_1+r_2}{2}\right) (r_2-r_1)\)

\(=2\pi\,r\,h \;\Delta r\)

where, \(\Delta r=r_2-r_1\)

\(\therefore\) Volume of cylindrical shell \(=2\pi\,r\,h \;\Delta r\)

The volume of solid obtained is

\(V=\displaystyle\int_a^b\,2\pi x\,f(x)\;dx\) where \(0\leq a <b\)

Cylindrical shell \(\rightarrow\;2\pi\,r\,h \;\Delta r\)

A \(\dfrac {2\,\pi}{7}\)

B \(5\,\pi\)

C \(\dfrac {62\,\pi}{5}\)

D \(\dfrac {52\,\pi}{3}\)

- Some volumes are difficult to calculate by the method of disk or washers.
- Consider the volume of solid obtained by rotating region bounded by \(y=0\), \(y=3x^2-x^3\) about y-axis.

- Since \(x\)is difficult to evaluate in terms of \(y\) , the method of disks is not very useful.
- Consider a cylindrical shell with inner radius \(r_1\) and outer radius \(r_2\).

Volume of shell = Volume of outer cylinder – Volume of inner cylinder

\(=V_2-V_1=\pi\,r_2^2\,h-\pi r_1^2\,h\)

\(=\pi\,h(r_2^2-r_1^2)\)

\(=\pi\,h(r_2+r_1)(r_2-r_1)\)

\(=2\pi\,h \left(\dfrac {r_1+r_2}{2}\right) (r_2-r_1)\)

\(=2\pi\,r\,h \;\Delta r\)

where \(\Delta r=r_2-r_1\)

\(\therefore\) Volume of cylindrical shell \(=\;\;2\pi\,r\,h \;\Delta r\)

The volume of solid obtained is

\(V=\displaystyle\int_a^b\,2\pi x\,f(x)\;dx\) where \(0\leq a <b\)

Cylindrical shell \(\rightarrow \;\;2\pi\,r\,h \;\Delta r\)

A \(5\,\pi\)

B \(8\,\pi\)

C \(\pi\)

D \(\dfrac {\pi}{24}\)