Q.Calculate consumer's surplus if the demand function $p=122-5x-2x^2$ and $x=20$.

At price $p_0=20$, $x_0=6$; $CS=\int_0^{6}p\,dx-p_0x_0=498-120=378$.

Here the given figure $20$ is taken as the market price $p_0=20$ (at $x=20$ the demand price $p=122-5x-2x^2$ would be negative, so $20$ must be the price at which the quantity demanded $x_0$ is found). Solve $122-5x-2x^2=20$:

$2x^2+5x-102=0\ \Rightarrow\ x=\frac{-5\pm\sqrt{25+816}}{4}=\frac{-5\pm 29}{4}\ \Rightarrow\ x_0=6.$

Consumer's surplus is

$CS=\int_0^{x_0}p\,dx-p_0x_0=\int_0^{6}(122-5x-2x^2)\,dx-(20)(6).$

…