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.

- The demand function \(P(x)\) is the price that a company has to charge in order to sell \(x\) units of a commodity. so selling larger quantities will require lowering price.

- The demand function is a decreasing function as \(x=\) number of units sold is increasing what P = price is decreasing.
- We define the consumer surplus = amount of money a consumer saves in purchasing the commodity at price P, corresponding amount demanded of X.

Consumer surplus = \(\int\limits_0^X(P(X)-P)\;dx\)

when, P = current selling price, where demand is X.

- The supply function \(P_s(x)\) of a commodity gives the relation between the selling price and number of units that manufacturer will produce at that price.
- \(P_s(x)\) is the increasing function of x, as manufacturer will produce more units if price is more.
- Some producer will sell the commodity for lower selling price and receive more than the minimal price. The excess is called producer surplus.

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

- The center of mass of any plate is the point P on which the plate balance horizontally.
- If we have system of n particles with masses \(m_1, m_2, m_3,.....m_n\) located on the points \(x_1, x_2, x_3,.....x_n\) on X-axis then the center of mass is located at \(\overline x=\dfrac {m_1x_1+m_2x_2+....+m_nx_n}{m_1+m_2+....+m_n}=\dfrac {\sum\limits_{i=1} ^nm_ix_i}{\sum\limits_{i=1}^{n}m_i}\)

for n = 2

- The value \(m_ix_i\) is called the moment of mass \(m_i\) about origin.
- \(M=\sum\limits_{i=1}^{n}m_ix_i\) is called moment of the system about the origin.
- If we consider the system of n particles with masses \(m_1,m_2....,m_n\) located at point \((x_1, y_1), (x_2, y_2),....,(x_n,y_n)\) in the X-Y plane. We define the moment of system about Y-axis as \(M_y=\sum\limits_{i=1}^nm_ix_i\) and moment of system about X axis to be \(M_x=\sum\limits_{i=1}^nm_iy_i\)

where, \(M_y=\) tendency of system to rotate about Y-axis

\(M_x=\) tendency of system to rotate about X-axis.

- \(\overline x=\dfrac {M_y}{m}, \) \(\overline y=\dfrac {M_x}{m}\), where \(m=\sum\limits_{i=1}^{n}m_i\)
- Center of mass = \((\overline x , \overline y)\)

- Consider a flat plate also called the lamina with uniform density \(\rho\)that occupies region R of the plane.

The center of mass or the centroid of this plate is given by \((\overline x, \overline y)\) when

\(\overline x =\dfrac {\int\limits_a^bxf(x) dx}{\int\limits_a^bf(x) dx}\), \(\overline y =\dfrac {\int\limits_a^b\,\dfrac{1}{2} (f(x))^2 dx} {\int\limits_a^bf(x) dx}\)

- If the region R lies between two curve \(y=f(x)\) and \(y=g(x)\) when \(f(x)\geq g(x)\)( \(f\)is higher than \(g\)), then

\(\overline x =\dfrac {\int\limits_a^bx\Big(f(x)-g(x)\Big) dx} {\int\limits_a^b\Big(f(x)-g(x)\Big) dx}\), \(\overline y =\dfrac {\int\limits_a^b\,\dfrac{1}{2} \Big((f(x))^2-(g(x))^2\Big) dx} {\int\limits_a^b\Big(f(x)-g(x)\Big) dx}\) 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 \((-1, 1)\)

- The theorem states that if R is a plane region that lies entirely on one side of line \(\ell\) and is rotated about \(\ell\), then volume of resulting solid is the product of area A of R and distance 'd' traveled by centroid of R.

A 320\(\pi^2m^3\)

B 160\(\pi^2m^3\)

C 2\(m^3\)

D 100000\(\pi^2m^3\)

- If the amount of capital that a company has at a time t is \(f(t)\) then the derivative \(f' (t)\) is called the net investment flow.

\(\therefore\) Capital formation function \(t_1\) to \(t_2\) \(=\int\limits_{t_1}^{t_2}\) \(f' (t) dt\)