Step 1

We have given a closed area which is enclosed by a line of equation $y = x$ and a curve of equation ${y^2} = 2x$.

We are asked to calculate the moment of inertia of the given area about the y-axis.

 
Step 2

Consider the equation of the line as:

\[{y_1} = x\]   ......(1)

Consider the equation of the given curve as:

\[\begin{array}{c} {y_2}^2 = 2x\\ {y_2} = \sqrt 2 {x^{\frac{1}{2}}} \end{array}\]   ......(2)
 
Step 3

Draw a differential element of the area parallel to the y-axis as shown below.

Here, $x$ is the distance of centroid of the differential element from the y-axis, ${y_2} - {y_1}$ is the length of the differential element, and $dx$ is the width of the differential element.

 
Step 4

The area of the differential element is given by:

\[dA = \left( {{y_2} - {y_1}} \right) \times dx\]

Substitute the equation (1) and (2) in the above equation:

\[dA = \left( {\sqrt 2 {x^{\frac{1}{2}}} - x} \right) \times dx\]   ......(3)
 
Step 5

The moment of inertia of the complete area about the y-axis is given by:

\[{I_y} = \int\limits_{0\;{\rm{m}}}^{2\;{\rm{m}}} {{x^2}dA} \]

Substitute the value of equation (3) in the above equation:

\[\begin{array}{c} {I_y} = \int\limits_{0\;{\rm{m}}}^{2\;{\rm{m}}} {{x^2}\left( {\left( {\sqrt 2 {x^{\frac{1}{2}}} - x} \right)dx} \right)} \\ = \int\limits_{0\;{\rm{m}}}^{2\;{\rm{m}}} {\left( {\sqrt 2 {x^{\frac{5}{2}}} - {x^3}} \right)dx} \\ = \left[ {\sqrt 2 \frac{{{x^{\frac{5}{2} + 1}}}}{{\frac{5}{2} + 1}} - \frac{{{x^{3 + 1}}}}{{3 + 1}}} \right]_{0\;{\rm{m}}}^{2\;{\rm{m}}}\\ = \left[ {\frac{{2\sqrt 2 {x^{\frac{7}{2}}}}}{7} - \frac{{{x^4}}}{4}} \right]_{0\;{\rm{m}}}^{2\;{\rm{m}}} \end{array}\]
 
Step 6

Apply the limits and solve the above equation:

\[\begin{array}{c} {I_y} = \left[ {\left( {\frac{{2\sqrt 2 {{\left( 2 \right)}^{\frac{7}{2}}}}}{7} - \frac{{{{\left( 2 \right)}^4}}}{4}} \right) - \left( {\frac{{2\sqrt 2 {{\left( 0 \right)}^{\frac{7}{2}}}}}{7} - \frac{{{{\left( 0 \right)}^4}}}{4}} \right)} \right]\;{{\rm{m}}^4}\\ = \left[ {\left( {4.571 - 4} \right) - \left( {0 - 0} \right)} \right]\;{{\rm{m}}^4}\\ = 0.571\;{{\rm{m}}^4} \end{array}\]