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You can skip the parallel axis theorem when the problem asks for the area moment of inertia directly about one of the coordinate axes used in the integral setup, provided there is no perpendicular offset distance to account for. In the specific example shown, the goal is to find $I_y$ about the y-axis itself.

Conditions: The requested axis is one of the coordinate axes (e.g., the x-axis or y-axis).; The integral is set up directly about that same axis.; No separate shifted-axis correction is being applied in the setup.

You can skip the parallel axis theorem when the problem asks for the area moment of inertia directly about one of the coordinate axes used in the integral setup, provided there is no perpendicular offset distance to account for. In the specific example shown, the goal is to find $I_y$ about the y-axis itself.

Conditions: The requested axis is one of the coordinate axes (e.g., the x-axis or y-axis).; The integral is set up directly about that same axis.; No separate shifted-axis correction is being applied in the setup.

The area moment of inertia about a specific axis is defined by integrating the square of the perpendicular distance from that axis over the entire area. For the y-axis, the perpendicular distance from any differential area element to the axis is the horizontal coordinate $x$.

Conditions: The moment is computed over a planar area in the xy-plane.; The axis of interest is either the x-axis or the y-axis.; The distance is measured perpendicularly from the axis to the differential area element.

The area moment of inertia about a specific axis is defined by integrating the square of the perpendicular distance from that axis over the entire area. For the y-axis, the perpendicular distance from any differential area element to the axis is the horizontal coordinate $x$.

Conditions: The moment is computed over a planar area in the xy-plane.; The axis of interest is either the x-axis or the y-axis.; The distance is measured perpendicularly from the axis to the differential area element.

The final numerical value of the area moment of inertia $I_y$ for the specific shaded region about the centroidal y-axis is approximately $0.762 \text{ m}^4$. This value is obtained by evaluating the definite integral $\int_0^2 x^2 (2 - \sqrt{2} x^{1/2}) dx$, which simplifies to $\frac{2^4}{3} - \sqrt{2} \left(\frac{2}{7}\right) 2^{3.5}$.

Conditions: The region is bounded by $y = 2$ (top) and $y = \sqrt{2} x^{1/2}$ (bottom) from $x=0$ to $x=2$.; The axis is the centroidal y-axis.; Units are in meters.

The final numerical value of the area moment of inertia $I_y$ for the specific shaded region about the centroidal y-axis is approximately $0.762 \text{ m}^4$. This value is obtained by evaluating the definite integral $\int_0^2 x^2 (2 - \sqrt{2} x^{1/2}) dx$, which simplifies to $\frac{2^4}{3} - \sqrt{2} \left(\frac{2}{7}\right) 2^{3.5}$.

Conditions: The region is bounded by $y = 2$ (top) and $y = \sqrt{2} x^{1/2}$ (bottom) from $x=0$ to $x=2$.; The axis is the centroidal y-axis.; Units are in meters.