5). Find the volume of the solid of revolution generated by rotating the graph of y=8 lnx about the y-axis between y=0 and y=16
Answer choices: 238.39pi, 246.39pi, 230.39pi, 214.39pi, 222.39pi
7). Find the area of the region bounded by y=√x-2, y=9, y=0, x=0
5. we will need x^2 for the square of the radius
y= 8lnx
y = ln x^8
or e^y = x^8
x^2 = e^(y/4)
V = π∫x^2 dy from 0 to 16
= π∫e^(y/4) dy from 0 to 16
= π [ 4 e^(y/4) ] from 0 to 16
= π ( 4e^4 - 4e^0)
= π (4e^4 - 4)
or
= 4π (e^4 - 1) or appr 214.39
5) To find the volume of the solid of revolution generated by rotating the graph of y=8 lnx about the y-axis, we can use the method of cylindrical shells.
First, let's find the points where the graph intersects the y-axis. Setting y=0 in the equation y=8 lnx gives us:
0 = 8 ln(x)
ln(x) = 0
x = 1
Therefore, we have a vertical line at x=1 that bounds the region we want to rotate.
Next, let's express the equation y=8 lnx in terms of x. Taking the exponential of both sides, we get:
e^(y/8) = x
Now, let's express the limits of integration in terms of y. We want to rotate the graph between y=0 and y=16. Using the equation e^(y/8) = x, we can find the corresponding x-values:
e^(16/8) = x
e^2 = x
Therefore, the limits of integration for y are 0 and 16, and the corresponding x-values are 1 and e^2.
Now, we can set up the integral to find the volume of the solid of revolution:
V = ∫[0,16] 2πx(y) dy
Since we have expressed x in terms of y, we can substitute it into the integral:
V = ∫[0,16] 2π(e^(y/8))(y) dy
Evaluating this integral will give us the volume of the solid of revolution.
To find the correct answer choice, you'll need to evaluate the integral on a calculator or by hand using integration techniques, such as integration by parts or substitution. The correct answer will be in the form of a numerical value multiplied by pi.
7) To find the area of the region bounded by y=√x-2, y=9, y=0, x=0, we can use the method of integration.
First, let's draw a graph of the given region and visualize it.
The region is bounded by the curves y=√x-2, y=9, y=0, and x=0. It lies between the x-axis and the line y=9.
To find the area of this region, we can set up the following integral:
A = ∫[0,81] (√x - 2) dx
The limits of integration for x are from x=0 to x=81, which are the x-values at which the curve y=√x-2 intersects y=9.
Evaluating this integral will give us the area of the region.
To find the correct answer choice, you'll need to evaluate the integral on a calculator or by hand using integration techniques, such as the power rule or u-substitution. The correct answer will be a numerical value.