Learn center of mass of centroid of complicated region & demand function. Practice to find the centroid of the region bounded by y & consumer surplus calculus.
Consumer surplus = \(\int\limits_0^X(P(X)-P)\;dx\)
when, P = current selling price, where demand is X.
Producer surplus = \(\int\limits_0^X(P-P_s(X))\; dx\)
where P is the price when, X items are sold.
A 11.25
B 22.5
C 500
D 1.25
for n = 2
where, \(M_y=\) tendency of system to rotate about Y-axis
\(M_x=\) tendency of system to rotate about X-axis.
The center of mass or the centroid of this plate is given by \((\overline x, \overline y)\) when
\(\overline x =\dfrac {\int\limits_a^bxf(x) dx}{\int\limits_a^bf(x) dx}\), \(\overline y =\dfrac {\int\limits_a^b\,\dfrac{1}{2} (f(x))^2 dx} {\int\limits_a^bf(x) dx}\)
\(\overline x =\dfrac {\int\limits_a^bx\Big(f(x)-g(x)\Big) dx} {\int\limits_a^b\Big(f(x)-g(x)\Big) dx}\), \(\overline y =\dfrac {\int\limits_a^b\,\dfrac{1}{2} \Big((f(x))^2-(g(x))^2\Big) dx} {\int\limits_a^b\Big(f(x)-g(x)\Big) dx}\) and centroid is \((\overline x, \overline y)\).
A \(\left ( \dfrac {11}{2}, \dfrac {5}{2\sqrt2} \right) \)
B \(\left ( \dfrac {24}{5}, \dfrac {3}{2\sqrt2} \right) \)
C \(\left ( 5, \dfrac {1}{\sqrt2} \right) \)
D \(\left ( -\dfrac {1}{2}, -\dfrac {1}{2} \right) \)
A \(\left ( \dfrac {5}{4}, \dfrac {3}{2} \right) \)
B \(\left (- \dfrac {1}{2}, 2 \right) \)
C \((6,8)\)
D \((-1, 1)\)
A 320\(\pi^2m^3\)
B 160\(\pi^2m^3\)
C 2\(m^3\)
D 100000\(\pi^2m^3\)
\(\therefore\) Capital formation function \(t_1\) to \(t_2\) \(=\int\limits_{t_1}^{t_2}\) \(f' (t) dt\)