PROBLEM 1: Compute the area of the enclosed region bounded by the graphs of the equations $y=x$, $y=2x$, and $x=4$.

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SOLUTION 1: Compute the area of the enclosed region bounded 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

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$x = 2x \ \ \ \ \longrightarrow$

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$- x = 0 \ \ \ \ \longrightarrow$

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$${-x} = 0 \ \ \ \ \longrightarrow$$

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$x = 0 \ \ \ \ \longrightarrow$

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Using vertical cross-sections we get that

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AREA $= \displaystyle{ \int_{0}^{4} (top \ - \ bottom) \ dx }$

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$= \displaystyle { \int_{0}^{4} (2x - x) \ dx }$

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$= \displaystyle { \int_{0}^{4} x \ dx }$

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$= \displaystyle { \frac{x^{2}}{2} \Big\vert_{0}^{4} }$

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$= \displaystyle { \frac{4^{2}}{2} - \frac{0^{2}}{2} }$

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$= \displaystyle { 8 - 0 }$

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$= \displaystyle { 8 }$