Solve the problem.
Find the volume of the solid generated by revolving the region bounded by the curve y=lnx, the x-axis, and the vertical line x=e^(2) about the x-axis.
One of the integrals you should have in your repertoire of common integrals is
∫lnx = xlnx - x
so volume
= π∫ lnx dx from x-1 to e^2
= xln - x | from 1 to e^2
= e^2(lne^2) - e^2 - (1ln1 - 1)
= e^2(2) - e^2 - 0 + 1
= e^2 + 1
I have these ans. chocies:
pi(e-1)
2pi(e^(2)-1)
pi(e^(2)-1)
pi e
Is it pi(e^(2)-1)
oops, sorry about dropping that π in the 4th last line
but, I don't see why it should not be
π(e^2 + 1)
check my arithmetic.
First, it's pi, not pie! Grrr! Does no math teacher ever teach that any more?
You can always check your answer using shells, instead of discs. As youi recall, the volume of a thin shell is essentially just the circumference times the height. So, since
y = lnx,
x = e^y
dx = e^y dy
and we have
v = ∫[0,2] 2πrh dy
v = 2π∫[0,2] y(e^2 - e^y) dy
= 2π(e^2/2 y^2 - ye^y + e^y) [0,2]
= 2π(e^2 - 1)
So, where did discs go wrong?
v = ∫[1,e^2] πr^2 dx
= π∫[1,e^2] (lnx)^2 dx
πx(ln^2(x)-2lnx+2) [1,e^2]
= πe^2(4-4+2) - π(0-0+2)
= 2π(e^2 - 1)
To find the volume of the solid generated by revolving the region bounded by the curve y=lnx, the x-axis, and the vertical line x=e^(2) about the x-axis, we can use the method of cylindrical shells.
1. First, let's visualize the region by plotting the curve y=lnx, the x-axis, and the vertical line x=e^(2). Based on the given information, we know that the region is bounded by x=e^(2), the x-axis, and the curve y=lnx.
2. Next, we need to determine the limits of integration. The left limit will be x=0 (since the curve starts from the origin) and the right limit will be x=e^(2) (the boundary given).
3. Now, let's consider a vertical strip at an arbitrary x-coordinate within the region. The height of this strip will be the difference between the x-axis and the curve y=lnx, which is h = x - ln(x).
4. To find the circumference of the cylindrical shell, we need to multiply the height by the circumference of the shell. The circumference of a shell is given by 2πr, where r is the distance of the strip from the axis of rotation. In this case, r is simply x since we are revolving about the x-axis.
5. The volume of each individual cylindrical shell is then given by V = 2πr * h.
6. Finally, we integrate the volume function from step 5 over the given limits of integration to obtain the total volume of the solid:
V = ∫(0 to e^(2)) 2πx * (x - ln(x)) dx
7. Evaluate the integral using appropriate techniques such as integration by parts or substitution.
8. Calculate the definite integral over the given limits of integration to get the final answer, which represents the volume of the solid generated.
Note: It's important to remember that the x-axis is the axis of rotation in this problem. If the y-axis were the axis of rotation, the method and equations used would be different.