Find the area of the region bounded by the curve $y\;=\;x^2\;,$ the x-axis, and the lines $x\;=\;1$ and $x\;=\;3.$

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Find the area of the region bounded by the curve $y\;=\;x^2\;,$ the x-axis, and the lines $x\;=\;1$ and $x\;=\;3.$

Find the area of the region bounded by the curve $y\;=\;x^2\;,$ the x-axis, and the lines $x\;=\;1$ and $x\;=\;3.$

Given the boundaries of the area to be found are,
• The curve $y\;=\;x^2$
• The $x-axis$
• $x\;=\;1$ (a line parallel to $y-axis$ )
• $x\;=\;3$ (a line parallel to $y-axis$ )

As per the given boundaries,
• The curve $y\;=\;x^2,$ has only the positive numbers as $x$ has even power, so it is about the $y-axis$ equally
distributed on both sides.
• $x\;=\;1$ and $x\;=\;3$ are parallel toy-axis at of $1$ and $3$ units respectively from the $y-axis.$
• The four boundaries of the region to be found are,
• $Point\;A,$ where the curve $y\;=\;x^2$ and $x\;=\;3$ meet
• $Point\;B,$ where the curve $y\;=\;x^2$ and $x\;=\;1$ meet
• $Point\;C,$ where the $x-axis$ and $x\;=\;1$ meet i.e. $C(1,0).$
• $Point\;D,$ where the $x-axis$ and $x\;=\;3$ meet i.e. $D(3,0).$
Area of the required region = Area of ABCD.