Finding Volume of Solid Revolved About Line y=4

In summary, the problem involves finding the volume of a solid generated by revolving the region enclosed by the graphs of f(x) = e^x and g(x) = ln(x) between x=1/2 and x=1 about the line y=4. The formula used is V= pi* integral( f(x)^2 - g(x)^2 dx). Another problem involves finding the time at which the number of people in an amusement park is maximum, given the rates at which people enter and leave the park. The approach is to find the maximum of H(t)= integral( E(x)-L(x)) from 9 to t for 9<=t<=23.
  • #1
nns91
301
1

Homework Statement



f(x) =e^x and g(x)= ln(x)

Find the volume of the solid generated when the region enclosed by the graphs of f and g between x=1/2 and x=1 is revolved about the line y=4

Homework Equations



v= pi* integral( f(x)^2 - g(x)^2 dx)

The Attempt at a Solution



SO for the part about the volume

I set up my integral as V= pi* integral ( (4-e^x)^2 - (4- ln(x))^2 dx) from 1/2 to 1 and get around -23. Am I right ? should it be 4-e^x or e^x-4 ?
 
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  • #2
Also another problem if you guys don't mind

The rate at which people enter an amusement park on a given day is modeled by the function E defined by E(t)= 15600/(t^2-24t+160)

The rate at which people leave is L(t)= 9890/(t^2-38t+370)

Both E(t) and L(t) are measured in people/hour an d time t is measured in hours after midnight. These functions are valid for 9<= x<= 23. At t=9, no one is in the park.

Let H(t)= integral( E(x)-L(x)) from 9 to t for 9<=t<=23. At what time t, for 9<=t<=23, does the model predict the number of people in the park is maximum.

My approach:

H(x) max when H'(x) min ( I am doubting this connection. AM I right ?). Then I find H''(x) and set it =0 and solve for t then plug in t back to the model. Is my approach right or should I do something else since I cannot t ( I keep getting negative numbers)
 

1. What is the formula for finding volume of solid revolved about line y=4?

The formula for finding the volume of a solid revolved about line y=4 is V = ∫π(R² - y²) dx, where R is the distance from the line y=4 to the outer edge of the solid.

2. How do you determine the limits of integration for finding volume of solid revolved about line y=4?

The limits of integration for finding volume of solid revolved about line y=4 are determined by the points where the solid intersects with the line y=4. These points will serve as the lower and upper limits of integration.

3. Can the volume of a solid revolved about line y=4 be negative?

No, the volume of a solid revolved about line y=4 cannot be negative. The volume represents the amount of space occupied by the solid, and it is always a positive value.

4. What is the difference between finding volume of solid revolved about line y=4 and finding volume of solid revolved about x or y axis?

The main difference is the axis of rotation. When finding volume of solid revolved about line y=4, the solid is rotated around a horizontal line, while finding volume of solid revolved about x or y axis involves rotating the solid around a vertical line.

5. Can the line y=4 be replaced with any other line when finding volume of solid revolved about line y=4?

Yes, the line y=4 can be replaced with any other horizontal line. This will change the axis of rotation and therefore, the formula for finding volume of solid revolved about that line will also change accordingly.

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