# Problem solving this volume using Jacobi's Determinant

• JorgeM
In summary, the problem is to find the volume of a solid with a ceiling defined by the equation 3x+4y+2z=10, cylindrical surfaces given by the equations 2x^2=y, x^2=3y, 4y^2=x, and y^2=3x, and the xy plane as the floor. The integral form for finding the volume is ∫∫z(x,y)dxdy, but it is easier to solve with a change of variables. The solid can be visualized as a pyramid with a blue line as the ceiling, crossing the z axis at 5 and indicated by the pink area. The limits for the integrals can be determined with this visualization
JorgeM

## Homework Statement

Find the value of the solid's volume given by the ecuation 3x+4y+2z=10 as ceiling,and the cilindric surfaces
2x^2=y
x^2=3*y
4y^2=x
y^2=3x
and the xy plane as floor.

## The Attempt at a Solution

I know that we have to give the ecuation this form:
∫∫z(x,y)dxdy= Volume
So, in fact we have to solve:
∫∫ ( 3 - 1.5x - .5y ) dxdy but actually it is easier to do variables' change because of the fact that the limits are to tricky to solve.

I get so confused when I try to suppose a good change and use in the Jacobi's determinant.
Hope you could help my because I got so confused :(

Just to visualize your solid. On the (x,y) plane it looks like the grey area in

where the blue line is from the ceiling, a pyramid which crosses the z axis at 5 indicated by the pink area.

You can try to figure out your limits for the integrals with this picture in mind.

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JorgeM

## What is Jacobi's Determinant and how is it used in problem solving?

Jacobi's Determinant is a mathematical formula used to find the determinant of a square matrix. It is often used in problem solving to solve systems of linear equations and to determine whether a matrix is invertible.

## What are the steps involved in solving a volume problem using Jacobi's Determinant?

The steps involved in solving a volume problem using Jacobi's Determinant include creating a matrix with the given variables, finding the determinant of the matrix, and using the value of the determinant to calculate the volume of the object in question.

## Can Jacobi's Determinant be used to solve volume problems in any shape?

Yes, Jacobi's Determinant can be used to solve volume problems in any shape as long as the dimensions of the object can be represented in a matrix form. It is a versatile formula that can be applied to various shapes and sizes.

## What are some common mistakes to avoid when using Jacobi's Determinant to solve volume problems?

Some common mistakes to avoid when using Jacobi's Determinant include incorrectly setting up the matrix, forgetting to find the absolute value of the determinant, and not considering the units of measurement when calculating the final volume.

## Are there any alternative methods to solving volume problems besides using Jacobi's Determinant?

Yes, there are other methods that can be used to solve volume problems, such as using integration or other determinants such as Cramer's Rule. However, Jacobi's Determinant is often preferred for its simplicity and efficiency in solving these types of problems.

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