Let R be the region bounded by y=x and y=x^2.Find the area of R.Find the volume of the solid that results when R is revolved about the x-axis(using the shell method). ​

Accepted Solution

Answer:2pi/15Step-by-step explanation:The intersection of y=x and y=x^2 is (0,0) and (1,1)Washer method Prefered this method for this one since slices are perpendicular to axis of rotationIntegrate(Big circle - Small Circle)dxIntegrate(pi*(x)^2-pi*(x^2)^2)dx on x=0..1Integrate(pi*x^2-pi*x^4)dx on x=0..1 is 2pi/15Shells methodSo I have to solve for x since I will be integrating with respect to yThe height of the the shell will be sqrt(y)-yThe radius will be y since that is the distance between axis of rotation in a point within the shellIntegrate(2*pi*r*h ) dyIntegrate(2*pi*(sqrt(y)-y)*y) dy on y=0..1 is 2pi/15