84. The base of a solid S is the region enclosed by the graph of y = (lnx)^(1/2), the line x = e, and thex-axis. If the cross sections of S perpendicular to the x-axis are squares, then the volume of S is
(A)12
(B)23
(C) 1
(D) 2
(E)13(e3-1)
84. The base of a solid S is the region enclosed by the graph of y = (lnx)^(1/2), the line x = e, and thex-axis. If the cross sections of S perpendicular to the x-axis are squares, then the volume of S is
(A)12
(B)23
(C) 1
(D) 2
(E)13(e3-1)
The formula is,Originally Posted by frozenflames
$\displaystyle \int^a_bA(x)dx$. Thus, you need to find a function for the area of each cross section $\displaystyle A(x)$. Since, the base is $\displaystyle \sqrt{\ln(x)}$ its height is also that because it forms square cross sections. Since, the area of the square is the sides squared we have $\displaystyle A(x)=(\sqrt{\ln x})^2=\ln x$. Now find the $\displaystyle a,b$ the problem says from the x-axis until x=e. To find the x-axis find when it is zero. Thus, $\displaystyle \sqrt{\ln x}=1$ which happens at $\displaystyle x=0$. Thus,
$\displaystyle V=\int^e_1\ln xdx=1$