Why this triple integral is not null?

In summary, the conversation discusses the concept of definite integration and the computation of volume using this technique. The disagreement arises when integrating sin(x) from 0 to pi, as one person argues that the integral should be 0 over a symmetric region while the other points out that the integration domain is not symmetric relative to zero. The difference between signed and unsigned areas is also mentioned. Ultimately, it is concluded that the area of a shape, such as a circle, is not always zero even if half of it is above the x-axis and half is below.
  • #1
Amaelle
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Homework Statement
look at the image
Relevant Equations
integrating impair function over symmetric region
Greetings
here is my integral
Compute the volume of the solid
1630172533382.png


and here is the solution (that I don't agree with)
1630172604791.png

1630172639981.png


So as you can see they started integrating sinx from 0 to pi and then multiplied everything by two! for me sin(x) is an odd function and it's integral should be 0 over symmetric region!
thank you in advance!
 
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  • #2
I am confused. Definite integration is a technique to find the area under the curve. Sin(x) is completely above the x-axis between (0, Pi). So the value isn't 0.
 
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  • #3
By definition, volume is positive and all its pieces have positive volume. So the fact that sin() is odd forces you to consider the negative part as an additional positive volume.
 
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  • #4
Amaelle said:
Homework Statement:: look at the image
Relevant Equations:: integrating impair function over symmetric region

So as you can see they started integrating sinx from 0 to pi and then multiplied everything by two! for me sin(x) is an odd function and it's integral should be 0 over symmetric region!
thank you in advance!
sin(x) is odd relative to zero. Your integration domain is not symmetric relative to zero. Hence, your argument does not apply.
 
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  • #5
Amaelle said:
So as you can see they started integrating sinx from 0 to pi and then multiplied everything by two! for me sin(x) is an odd function and it's integral should be 0 over symmetric region!
thank you in advance!
The area of (say) a circle, centred somewhere on the x-axis, is not zero - even though half the area is above the x-axis and half is below.

Beware of the difference between signed and unsigned areas!
 
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  • #6
Steve4Physics said:
The area of (say) a circle, centred somewhere on the x-axis, is not zero - even though half the area is above the x-axis and half is below.

Beware of the difference between signed and unsigned areas!
thank you I got it!
 
  • #7
Steve4Physics said:
The area of (say) a circle, centred somewhere on the x-axis, is not zero - even though half the area is above the x-axis and half is below.

Beware of the difference between signed and unsigned areas!
Orodruin said:
sin(x) is odd relative to zero. Your integration domain is not symmetric relative to zero. Hence, your argument does not apply.
thank you very much! I understood the difference now!
 
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FAQ: Why this triple integral is not null?

Why is the triple integral not null?

The triple integral is not null because it represents the volume under a three-dimensional surface or within a three-dimensional region. This means that there is a non-zero amount of space enclosed by the surface or region, resulting in a non-zero value for the integral.

Can the triple integral ever be null?

In some cases, the triple integral can be equal to zero. This occurs when the function being integrated is equal to zero over the entire region of integration. This means that the volume under the surface or within the region is completely cancelled out, resulting in a null value for the integral.

What factors can affect the value of the triple integral?

The value of the triple integral can be affected by the function being integrated, the limits of integration, and the shape and size of the region being integrated over. Additionally, the method of integration and the accuracy of the calculations can also impact the final value.

How can I determine if a triple integral will be null or non-null?

To determine if a triple integral will be null or non-null, you can evaluate the function being integrated over the given region and check if it is equal to zero. If the function is not equal to zero, then the integral will be non-null. Additionally, you can also visualize the region and the function to get a better understanding of the potential volume enclosed.

Are there any real-world applications for triple integrals?

Yes, triple integrals have many real-world applications in fields such as physics, engineering, and economics. They can be used to calculate the volume of three-dimensional objects, the mass of a three-dimensional object with varying density, and the center of mass of a three-dimensional object. They can also be used to solve optimization problems and calculate probabilities in statistics.

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