SOLUTION 1: Compute the area of the region enclosed by the graphs of the equations $ y=x $, $ y=2x $ and $ x=4 $ . Begin by finding the points of intersection of the two graphs. From $ y=x $ and $ y=2x $ we get that $$ x = 2x \ \ \longrightarrow \ \ x = 0 $$ Now see the given graph of the enclosed region.

Using vertical cross-sections to describe this region, we get that $$ 0 \le x \le 4 \ \ and \ \ x \le y \le 2x $$ so that the area of this region is $$ AREA = \displaystyle{ \int_{0}^{4} (Top \ - \ Bottom) \ dx } $$ $$ = \displaystyle { \int_{0}^{4} (2x - x) \ dx } $$ $$ = \displaystyle { \int_{0}^{4} x \ dx } $$ $$ = \displaystyle { \frac{x^{2}}{2} \Big\vert_{0}^{4} } $$ $$ = \displaystyle { \frac{4^{2}}{2} - \frac{0^{2}}{2} } $$ $$ = \displaystyle { 8 - 0 } $$ $$ = \displaystyle { 8 } $$

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