Find the area of the region bounded by the parabola $y^2\;=\;4x,$ the $x-axis,$ and the lines $x\;=\;1$ and $x\;=\;4.$

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Find the area of the region bounded by the parabola $y^2\;=\;4x,$ the $x-axis,$ and the lines $x\;=\;1$ and $x\;=\;4.$

Find the area of the region bounded by the parabola $y^2\;=\;4x,$ the $x-axis,$ and the lines $x\;=\;1$ and $x\;=\;4.$

Given the boundaries of the area to be found are,
• The parabola $y^2\;=\;4x$
• The $x-axis$
• $x\;=\;1$ (a line parallel to $y-axis$ )
• $x\;=\;4$ (a line parallel to $y-axis$ )

As per the given boundaries,
• The curve $y^2\;=\;4x,$ has only the positive numbers as $y$ has even power, so it is about the $x-axis$ equally
distributed on both sides.
• $x\;=\;1$ and $x\;=\;4$ are parallel to $y-axis$ at of $1$ and $4$ units respectively from the $y-axis.$
• The four boundaries of the region to be found are,
• $Point\;A,$ where the curve $y^2\;=\;4x,$ and $x\;=\;4$ meet
• $Point\;B,$ where the curve $y^2\;=\;4x,$ 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 $x\;=\;4$ meet i.e. $D(4,0).$
Area of the required region = Area of ABCD