Question

R is the first quadrant region enclosed by the x-axis and the curve y = 2x + b, and the line x = b, where b > 0. Find the value of b so that the area of the region R is 288 square units

Answer 1

The area of $R$ is given by the integral

$\displaystyle\int_0^b(2x+b)\,\mathrm dx=288$

Integrating yields

$(x^2+bx)\bigg|_{x=0}^{x=b}=(b^2+b^2)-(0+0)=2b^2$

So you have

$2b^2=288\implies b^2=144\impiles b=12$