Question

In Exercises \(41-44\), set up and evaluate the integrals for finding the area and moments about the \(x\) - and y-axes for the region bounded by the graphs of the equations. (Assume \(\rho=1\).) $$ y=x^{2}, y=x $$

Short answer

The area of the region is \(A = 0.1667\) square units. The moments around the x- and y-axes are \(M_x = -0.05\) and \(M_y = 0.25\) respectively.

Step-by-step solution

Find common points

To find the bounds for the area integral, solve \(x = x^2\). The solutions yield the points (0,0) and (1,1).

Compute the area

The area \(A\) of the region is given by the integral \(\int_a^b f(x) - g(x))dx\), where \(f(x) = x\) and \(g(x) = x^2\). Here, using the limits from step 1, \(a = 0\) and \(b = 1\). Therefore, \(A = \int_0^1 (x - x^2)dx\). Evaluating this, we find that the area is \(A = 0.1667\) square units.

Compute the moments about x and y axes

The moments about the x-axis \(M_x\) and y-axis \(M_y\) are given by \(-\int_a^b f(x)*g(x)dx\) and \(\int_a^b x*f(x)dx\) respectively. Evaluating these integrals, we obtain \(M_x = -0.05\) and \(M_y = 0.25\).