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Emekadavid

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thanks

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In summary: It is just a matter of applying the concept correctly in different situations. In summary, double integrals can indeed be interpreted as net change in volume, and the use of dA and Riemann's sums can help us understand this concept better.

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Emekadavid

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thanks

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- #2

jvicens

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I can clarify the intuitive understanding of double integrals for you. Double integration can indeed be interpreted as net change, but in this case, it represents the net change in volume. This is because double integrals are used to find the volume under a surface in three-dimensional space.

To better understand this, let's first consider the single integral. When we integrate a function over a certain interval, we are essentially finding the net change in that function over that interval. For example, if we have a function that represents the velocity of an object, integrating it over a certain time interval would give us the net change in the position of the object over that time interval.

Similarly, in double integration, we are finding the net change in volume under a surface. This can be thought of as the net change in the quantity represented by the surface, such as the amount of water in a container or the amount of air in a room.

To answer your question about whether double integration can be regarded as net change when one variable is held constant and the other is changing, the answer is yes. This is because in this case, we are essentially looking at the net change in volume as one variable changes, while the other remains constant.

The use of dA in double integrals represents the infinitesimal area of the surface, which is then multiplied by the function f(x,y) to find the volume under that small area. By summing up all these infinitesimal volumes, we can approximate the total volume under the surface.

I hope this clarifies the intuitive understanding of double integrals for you. Remember, in the end, it all comes down to finding the net change in volume under a surface, whether both variables are changing or one is held constant.

A double integral is a mathematical concept that allows us to find the volume under a surface in three-dimensional space. It is essentially a way to calculate the net change of a function over a two-dimensional region.

A double integral can be interpreted as the net change of a function over a two-dimensional region. This is because the integral calculates the sum of infinitely small changes in the function over the given region, resulting in the overall net change.

Yes, double integrals can be used to find net change in various real-world scenarios, such as calculating the total mass of an object or the total amount of fluid flowing through a pipe. It is a useful tool in physics, engineering, and other scientific fields.

A single integral calculates the net change of a function over a one-dimensional interval, while a double integral calculates the net change over a two-dimensional region. Essentially, a single integral is a special case of a double integral.

Double integrals may not be applicable in certain scenarios where the function being integrated is discontinuous or undefined over the given region. In these cases, alternative methods such as using line integrals may be necessary to calculate net change.

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