A curve is such that the area of the region b | vedclass

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(C) Let the curve be $y = f(x)$. The area of the region bounded by the coordinate axes,the curve,and the ordinate at point $(x, y)$ is given by $\int_

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a parabola

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(C) Let the curve be $y = f(x)$. The area of the region bounded by the coordinate axes,the curve,and the ordinate at point $(x, y)$ is given by $\int_0^x y \, dx = y^3$.Differentiating both sides with respect to $x$ using the Leibniz rule:$y = \frac{d}{dx}(y^3)$$y = 3y^2 \frac{dy}{dx}$This gives two cases:$1) y = 0$ (which is a trivial solution,the x-axis).$2) 1 = 3y \frac{dy}{dx} \Rightarrow 3y \, dy = dx$.Integrating both sides:$\int 3y \, dy = \int dx$$\frac{3y^2}{2} = x + C$Rearranging the equation:$y^2 = \frac{2}{3}(x + C)$This is the standard form of a parabola $y^2 = 4a(x - h)$.

$A$ curve is such that the area of the region bounded by the coordinate axes,the curve,and the ordinate of any point on it is equal to the cube of that ordinate. The curve represents:

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