Find Mass, Mx, My and x̅ and y̅ of the Lamina bounded by R =...

Find Mass, Mx, My and \(\bar{x}\) and \(\bar{y}\) of the Lamina bounded by R = \(\{(x, y) \mid 0 \le x \le 5, -1 \le y \le 1\}\) and with the density function \(\rho(x, y) = 5 \cdot x \cdot y^2\). Provide an answer accurate to at least significant digits.

Transcript

00:01 Hello everyone, we need to find the mass mx, m y, x bar and y bar of the given lamina.

00:19 So we are given with the equation y is equal to minus 8x plus 3 where y is equal to x by 2 and x is equal to 0 so also we are given with the density row of x comma y is equal to 3 times of x plus y.

00:43 So let us equate this y is equal to minus 8x plus 3 and substitute x by 2 for y.

00:52 We would get minus 8x plus 3 is equal to x by 2 or we get minus 16 x plus 6 is equal to x.

01:05 From this we obtain 6 is equal to 17x or this is nothing but x is equal to 6 by 17.

01:14 So let us first compute the mass which is given by the formula.

01:19 Mass is equal to double integral over r row of x comma y, d y, d x.

01:29 So let us substitute the limits, that is, mass is equal to integral 0 to 6 by 17, integral x by 2 to minus 8x plus 3, 3 times x plus y, d, dx.

01:52 So simplifying this, we obtain integral 0 to 6 by 17, 3xyxy, plus 3xy plus 3y plus 3y, plus 3y, and x.

02:01 Square by 2 with respect to the limits x by 2 to minus 8x plus 3 dx.

02:11 So evaluating this integral, the value of mass will be approximately equal to 1 .8685.

02:22 Now we need to compute the mx value, that is m .x is equal to double integral over r.

02:35 Row of x comma y y d y d x so the limits will be same as we have computed for mass so this will be equal to integral 0 to 6 by 17 integral x by 2 to minus 8x plus 3 3 times x plus y into y d y d y d x so similar manner we will evaluate the that is integral 0 to 6 by 17 integral x by 2 to minus 8x plus 3 xy plus 3y plus 3y square dy d x simplifying further we would get integral 0 to 6 by 17 3xxy xy square by 2 plus 3y y square by 2 plus 3y cube by 3 with respect to the limits x by 2 to minus 8x plus 3 d x.

03:50 So evaluating this entire integral we would get the value of mx to be approximately equal to 2 .6874...

Question

Find Mass, Mx, My and \(\bar{x}\) and \(\bar{y}\) of the Lamina bounded by R = \(\{(x, y) \mid 0 \le x \le 5, -1 \le y \le 1\}\) and with the density function \(\rho(x, y) = 5 \cdot x \cdot y^2\). Provide an answer accurate to at least significant digits.

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