The area of the parabola y = x2 from (1,1) to (2,4) is rotated about t | Mathematics I 2080

Mathematics I208010 marksNumerical: surface area and ODEArea of surface of revolution

The area of the parabola $y = x^2$ from (1,1) to (2,4) is rotated about the y-axis. Find the area of the resulting surface. Find the solution of the equation $y^2 dy = x^2 dx$ that satisfies the initial condition $y(0) = 2$ . [5+5]

(a) Surface Area of Revolution of $y = x^2$ about the Y-axis - Curve: $y = x^2$ - Segment from $(1,1)$ to $(2,4)$ - Axis of rotation: y-axis For rotation about the y-axis: $S = 2\pi \int x \, ds, \quad ds = \sqrt{1 + \left(\frac{dx}{dy}\right)^2}\, dy$ From $y = x^2$ : $x = \sqrt{y}$ , so $$\frac...

(a) Surface Area of Revolution of $y = x^2$ about the Y-axis - Curve: $y = x^2$ - Segment from $(1,1)$ to $(2,4)$ - Axis of rotation: y-axis For rotation about the y-axis: $$S = 2\pi \int x , ds, \quad ds = \sqrt{1 + \left(\frac{dx}{dy}\right)^2}, dy$$ From $y = x^2$: $x = \sqrt{y}$, so $$\frac...

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The area of the parabola y = x2 from (1,1) to (2,4) is rotated about the y-axis. Find the area of the resulting surface. Find the solution of the equation y2 dy = x2 dx that satisfies the initial condition y(0) = 2. [5+5]