I'm having trouble calculating the volume of a vase. Moreover, i've found the equation and found the volume through one method - in which i divided the vase into 100 cylinders. I summed the 100 volumes of the 100 cylinders and ended with 1509.192376cm^3.
I'm asking for help to find the volume of the vase using calculus as I need to validate whether my cylindrical method is accurate is works.In regards to calculus, i was thinking of using integration but can't seem to put my head around it...
The equation: Y = 0.00007202x^6 - 0.00298409x^5 + 0.04474982x^4 - 0.28787259x^3 + 0.68195292x^2 - 0.10780426x + 6.03715171
For integration: a = 0, b = 12.
**I have called the function a vase, as i am suppose to imagine that the function would be flipped onto the other side as a reflection. Also, i know it still wouldnt actually represent a vase, but it kinda does i guess?
I was not given an equation for the function, I was instead given x and y co-ordinates (see below) in which I used in excel to create a graphical representation. x y 0 6 1 6.5 2 6.75 3 7 4 6.75 5 6.5 6 6 7 5.5 8 5.75 9 6.25 10 6.5 11 6.25 12 6
Additionally, I was instructed to choose a polynomial 6 trendline as it had the best r^2 value.
The volume of a curve rotated about the $x$-axis is well known through variuos methods (e.g., disc and Pappus's $2^{nd}$ Centroid theorem) and is given by
$$V=\pi \int y^2~dx$$
In your case we find
$$V=\pi \int_0^{12} y^2~dx\approx 1509.42$$