# Definite integration

2021-12-26
The solid lies  between the Planes perpendicular to the x axis at X=-1amd X=1. Cross-sections perpendicular to the x axis between between these planes are circular disk with the diameters run from the semicircle y =-root(1-x*2)to the semi circle y=root(1-x*2).Find a formula for the area of cross section A(x)

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karton

The bounds of the region are x -1 and x =1 so integrate the area function from -1 to 1 to find the volume.

$$\begin{array}{} V=\int^1_{-1}A(x)dx\\ V=\int^1_{-1} 4(1-x^2)dx\\ =4(x-\frac{1}{3}x^3)|^1_{-1}\\ =4(1-\frac{1}{3}(1)^3)-4(-1-\frac{1}{3}(-1)^3)\\ =4(1-\frac{1}{3})-4(-1+\frac{1}{3})\\ =4(\frac{2}{3})-4(-\frac{2}{3})\\ =\frac{8}{3}+\frac{8}{3}\\ =\frac{16}{3} \end{array}$$