1. The figure below shows a sketch of the curve with equation $y=1+2x-\frac{1}{4}{x}^{2}$ where

• the point  $P(x,y)$ lies on the curve

• the shaded rectangle shown has width $\delta x$ and height $y$

By expressing the series limit as a suitable integral, show that

$\underset{ \delta x\rightarrow0}{lim}\sum_{x=1}^{8}(1+2x-\frac{1}{4}{x}^{2}) \delta x=\frac{329}{12}$