Find the volume of revolution about the x-axis of the curve y=1/sqrt(x^2+2x+2) for 0<x<1
The volume of revolution is given by integrating Piy2 dx from 0 to 1.Squaring, y2=1/(x2+2x+2)Completing the square, we see that y=1/((x+1)2+1)Make the substitution u=x+1, so du=dx. When x is 0, respectively 1, u is 1, respectively 2. So the volume is the integral of Pi/(u2+1) du from 1 to 2. This is Piarctan(u) evaluated from 1 to 2, which is Pi*(arctan(2)-arctan(1)). In a calculator, we see this is roughly 1.011 and this is the desired volume.
HG
Answered by Harry G. • Further Mathematics tutor
3027 Views
See similar Further Mathematics A Level tutorsRelated Further Mathematics A Level answers
All answers ▸find general solution to: x(dy/dx) + 2y = 4x^2
Further Maths: How do you find the inverse of a 2 x 2 matrix?
For a homogeneous second order differential equation, why does a complex conjugate pair solution (m+in and m-in) to the auxiliary equation result in the complementary function y(x)=e^(mx)(Acos(nx)+Bisin(nx)), where i represents √(-1).
Prove by induction that n^3+5n is divisible by 3 for every natural number.