A thin plate covers the triangular region bounded by the x-axis and the lines x = 1 and y = 2x in the first quadrant. The plate's density at the point (x, y) is δ(x, y) = 6x + 6y +... A thin plate covers the triangular region bounded by the x-axis and the lines x = 1 and y = 2x in the first quadrant. The plate's density at the point (x, y) is δ(x, y) = 6x + 6y + 6. Find the plate’s moments of inertia about the coordinate axes and the origin. Read more

Steps to Solve

1. Determine the Region of Integration

The triangular region is bounded by:

• The x-axis (y = 0)
• The line $x = 1$
• The line $y = 2x$

The vertices of the triangle are (0,0), (1,0), and (1,2).

2. Set up the Density Function

Given the density function: $$ \delta(x, y) = 6x + 6y + 6 $$

3. Define the Moments of Inertia Formulas

The moments of inertia of a thin plate about the x-axis, y-axis, and origin are given by:

• About the x-axis: $$ I_x = \int \int y^2 \delta(x,y) , dA $$
• About the y-axis: $$ I_y = \int \int x^2 \delta(x,y) , dA $$
• About the origin: $$ I_O = \int \int (x^2 + y^2) \delta(x,y) , dA $$

4. Set Up the Double Integrals

For the triangular region, we'll integrate over the bounds for $x$ from 0 to 1 and for $y$ from 0 to $2x$:

• Moment about the x-axis: $$ I_x = \int_0^1 \int_0^{2x} y^2 (6x + 6y + 6) , dy , dx $$

• Moment about the y-axis: $$ I_y = \int_0^1 \int_0^{2x} x^2 (6x + 6y + 6) , dy , dx $$

• Moment about the origin: $$ I_O = \int_0^1 \int_0^{2x} (x^2 + y^2) (6x + 6y + 6) , dy , dx $$

5. Calculate $I_x$

First, calculate the inner integral for $I_x$: $$ I_x = \int_0^1 \left( \int_0^{2x} y^2 (6x + 6y + 6) , dy \right) dx $$

Evaluate the inner integral by expanding and integrating:

$$ \int_0^{2x} y^2 (6x + 6y + 6) , dy = 6x \int_0^{2x} y^2 , dy + 6 \int_0^{2x} y^3 , dy + 6 \int_0^{2x} dy $$

Compute these integral forms:

• $ \int y^2 , dy = \frac{y^3}{3} \Bigg|_0^{2x} = \frac{(2x)^3}{3} = \frac{8x^3}{3} $
• $ \int y^3 , dy = \frac{y^4}{4} \Bigg|_0^{2x} = \frac{(2x)^4}{4} = \frac{16x^4}{4} = 4x^4 $
• $ \int dy = y \Bigg|_0^{2x} = 2x $

Substituting these back will yield the integral for $I_x$. Then, evaluate the outer integral over $x$.

6. Calculate $I_y$ and $I_O$

Follow a similar process for $I_y$ and $I_O$ using their respective integrals and density function.

7. Final Computation

Collect the results after evaluating all three integrals.