The area of the region is bounded by the curve [tex]\rm y=e^2x[/tex] the x-axis the y axis, and the line x=2 is equal to [tex]\rm \dfrac{e^4}{2}-\dfrac{1}{2}[/tex].
Given that,
The area of the region bounded by the curve [tex]\rm y=e^2x[/tex],
We have to determine,
The x-axis the y axis and the line x=2 is equal to.
According to the question,
The area of the region bounded by the curve
[tex]\rm y=e^2x[/tex]
The area of the region bounded by the curve is determined by integrating the curve at x = 0 to x = 2.
Integrating the curve on both sides,
[tex]\rm Area=\int\limits^2_0 { e^{2x}} \, dx\\\\Area=[ \dfrac{e^{2x}}{2}]^2_0\\\\Area= [ \dfrac{e^{2(2)}}{2}- \dfrac{e^{2(0)}}{2}]\\\\Area = \dfrac{e^4}{2}-\dfrac{e^0}{2}\\\\Area = \dfrac{e^4}{2}-\dfrac{1}{2}[/tex]
Hence, The area of the region is bounded by the curve [tex]\rm y=e^2x[/tex] the x-axis the y axis, and the line x=2 is equal to [tex]\rm \dfrac{e^4}{2}-\dfrac{1}{2}[/tex].
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