Approximate Integration (1 Viewer)

[xwrathbringerx]

use simpson's rule with five function values to estimate the volume of the solid formed by rotating the curve y = 1/(1+x^2) about the X-axis between x = -2 and x=2.

I've never seen this sort of question before, combining both volumes and area under curve. Could someone please show me how it is done?<!-- google_ad_section_end -->

 

[tommykins]

apply simpsons rule to the expression for the volume of the graph

 

[Drongoski]

use simpson's rule with five function values to estimate the volume of the solid formed by rotating the curve y = 1/(1+x^2) about the X-axis between x = -2 and x=2.

I've never seen this sort of question before, combining both volumes and area under curve. Could someone please show me how it is done?<!-- google_ad_section_end -->

You can do it in 2 parts:

i) V = pi x integral (-2 to 2) y² dx

ii) but you can't do the integration; so you use Simpson's rule to give you a numerical approxn. So you calculate the values of y^2 @ the 5 equally-spaced points viz. x = -2, -1, 0, 1 , 2 . Call these 5 values Y1, Y2, Y3, Y4, Y5

Then: pi x integral [-2 to 2] of y^2 dx is approximated by

V = pi x h/3[1.Y1 + 4.Y2 + 2.Y3 + 4.Y4 + 1.Y5] where h = 1

 

[bored.of.u]