V Solutiony1x yx and x3 1xx xx 1 x321 x1 1

Solution

y=(1/x), y=x, and x=3.

(1/x)=x

xx =1

x^3/2=1

x=1

1<=x<=3, 1_/x<=y<=x

volume under the surface,V =_{[1 to 3][1/x tox]}(x+2xy) dy dx

V =_{[1 to 3][1/x tox]}(xy+xy²) dx

V =_{[1 to 3]}(xx+x(x)²)-(x(1_/x)+x(1_/x)²) dx

V =_{[1 to 3]}(x^3/2+x²)-(1+(1_/x)) dx

V =_{[1 to 3]}((2_/5)x^5/2+(1_/3)x³)-(x+ln(x))

V=((2_/5)3^5/2+(1_/3)3³)-(3+ln(3)) -((2_/5)1^5/2+(1_/3)1³)+(1+ln(1))

V=(183_/5) +9-3-ln(3) -(2_/5)-(1_/3)+1+0

V=(183_/5)-ln(3)  +(94_/15) 11.4034