The slope of the tangent to the curve \(y=f(x)\) at \((x,f(x))\) is \(2x\). If the curve passes through the origin, then what is the area bounded by the curve, the x-axis and the line \(x=1\)?
1/3 square unit
Since \(\dfrac{dy}{dx}=2x\), integrating gives \(y=x^{2}+C\); as the curve passes through the origin, \(C=0\), so \(y=x^{2}\). The area bounded by this curve, the x-axis and \(x=1\) is \(\displaystyle\int_{0}^{1}x^{2}\,dx=\dfrac{1}{3}\) square unit.
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Where do the curves intersect?
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