Question

Find Mx, My, and for the laminas of uniform density ρ bounded by the graphs of the equations.

y = x2 , y = x3

Answer 1

Step 1:

The equations of the curves are  and  .

Determine the points of intersection by plotting their graphs.

Graph:

Draw the coordinate plane.

Graph the curves and .

Observe the graph, the two curves intersect at the points and .

Step 2:

Consider and .

Moment about - axis of a planar lamina is defined as,

Where, uniform density of a planar lamina.

Here value varies form 0 to 1, so and .

Substitute , and limits of integral in the above formula.

Answer 2

Continued:

Step 3:

Moment about - axis of a planar lamina is defined as,

Substitute corresponding values in the formula.

Step 4:

Center of mass  and .

Mass of the system is .

Substitute and in the above formula.

Mass of the system is

Substitute , and values in center of mass formula.

Center of mass .

Solution:

Moments and center of mass .