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How would the following equation be used to find an equation for the surface area of the 3d shape?

$$y=0.7\sqrt{6.5^{2}-x^{2}}\left(0.05x+0.9\right)$$

The extension sheet described it as half of a vertical cross section of an asymmetric oval (so basically a half-ellipse). The question wanted a function of the surface area since in later questions it will be needed to calculate the object's volume.

From what I could find online, the integral of an ellipse's circumference gives you the surface area. Since the aforementioned equation is basically the radius of the object, I tried to just multiply it by 2 $\pi$ but currently this is where I am stuck.

I'm not sure if the question just wants the integration of $$y=\left(0.05x+0.8595\right)\cdot0.7\sqrt{6.5^{2}-x^{2}}\cdot2\pi$$ as the function for the oval's surface?

I also graphed it on plots to try and visualise how there might be some other ways of equation revolving that is needed but currently I feel very out of my depth.

Help would be very appreciated and sorry for the wordy nature of all the descriptions, the original question's wording is confusing me too.

Thomas Andrews
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    See: https://math.stackexchange.com/questions/1691147/why-is-surface-area-not-simply-2-pi-int-ab-y-dx-instead-of-2-pi-in – Thomas Andrews May 03 '24 at 11:16
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    But the equivalent would be to estimate the length of a curve between $a,b$ as $\int_a^b dx,$ ignoring the slope of the function at each point, just estimating the curve's length by the horizontal line segments. – Thomas Andrews May 03 '24 at 11:18
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    Look up the surface of revolution – Paul May 03 '24 at 11:24
  • The y= f(x) equation doesn't describe a 3D shape. Please clarify what surface you are trying to calculate the area of. – Simon Goater May 03 '24 at 11:59

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