Definite integral distance from volume

In summary, the conversation is about finding the distance from the bottom of a spherical water tower to the water level if it is filled to 1/4 of its capacity. The solution involves setting up and solving a definite integral, and using numerical techniques such as Newton's method or graphing to find the answer.
  • #1
poopcaboose
33
0

Homework Statement

If you have a water tower that is spherical with a radius 20m, how far from the bottom will the water level be if it is filled to 1/4 of it's capacity.



2. Homework Equations [tex]\int_-20^20pi(400-y^2)dy=33510.32164m^3=3,351,032.164Liters[/tex]



The Attempt at a Solution

I can find 1/4 capacity=837,758.041 liters but don't know where to go from here to find the distance from the bottom
 
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  • #2
I meant for it to be a definite integral from 20 to -20 not 20 squared
 
  • #3
Instead of integrating from -20 to 20 to get the full volume, integrate from -20 to h. You'll get a cubic expression in h. Equate that to 1/4 of the full volume and try to solve for h.
 
  • #4
BTW it looks like you have to use numerical techniques to get an answer. The cubic doesn't factor or anything for me.
 
  • #5
Thanks for the help, I can use Newtons method to approximate f(y) at zero and solve it from there thanks I would have never thought to set it to y
 
  • #6
By the way, two of the roots are complex numbers from what I got.. just a warning.
 
  • #7
I figured it out its 13.054073m from the bottom you either solve for zero with Newtons method or graph it
 

FAQ: Definite integral distance from volume

What is a definite integral distance from volume?

A definite integral distance from volume refers to the distance between the x-axis and the curve of a function at a specific interval. This distance is calculated using the definite integral formula, which represents the area under the curve between the two points on the x-axis. It is often used in physics and engineering to calculate the volume of irregularly shaped objects.

How is the definite integral distance from volume calculated?

The definite integral distance from volume is calculated by taking the definite integral of a function between two points on the x-axis. This means finding the area under the curve of the function between these two points. The definite integral formula involves taking the limit of a sum of infinitely small rectangles under the curve, which gives an accurate measurement of the distance from volume.

What are the applications of definite integral distance from volume?

Definite integral distance from volume has many applications in physics and engineering. It is used to calculate the volume of irregularly shaped objects, such as tanks, pipes, and containers. It is also useful in calculating the amount of fluid or gas that can be contained within a certain space. In addition, it is used in the study of fluid dynamics, where it helps determine the flow rate and pressure of fluids.

What is the relationship between definite integral distance from volume and the volume of an object?

The definite integral distance from volume is directly related to the volume of an object. This is because the definite integral formula calculates the area under the curve, which represents the volume of the object. The larger the area under the curve, the larger the volume of the object. Therefore, by calculating the definite integral distance from volume, we can accurately determine the volume of an irregularly shaped object.

Can definite integral distance from volume be negative?

Yes, definite integral distance from volume can be negative. This occurs when the curve of a function dips below the x-axis, resulting in a negative area under the curve. In this case, the definite integral distance from volume represents the distance below the x-axis, which is equivalent to the negative volume of the object in that specific interval.

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