Volume of Solid Multivariable Calc

The xy-plane is a 2D space and can be represented by the unit circle. The bounds for the double integral would be [-1,1] for both x and y since it is restricted by the unit circle. In summary, the volume of the solid in the first octant of xyz space is bounded below by the coordinate axes and the unit circle and bounded about by z = 8xy can be found by evaluating the double integral of 8xy with limits of integration [-1,1] for both x and y.
  • #1
l.daniels241
1
0
Find the volume of the solid in the first octant of xyz space, bounded below by the coordinate axes and the unit circle and bounded about by z = 8xy

A) 1/2
B) 1
C) 2
D) 4
E) 8

I know we need a double integral. The bound below should be the unit circle which would be
x^2 + y^2 = 1. So x goes from -1 to 1 i think.

I know it should be the double integral of 8xy but i do not know my limits of integration i am not sure how to find that out...

Can someone help find the limits please
 
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  • #2
Have you considered looking in your calculus book in the double integral section to review how to get the limits?
 

Related to Volume of Solid Multivariable Calc

1. What is the formula for calculating the volume of a solid in multivariable calculus?

The formula for calculating the volume of a solid in multivariable calculus is given by the triple integral, which takes into account the three dimensions of length, width, and height. The integral is written as ∭f(x,y,z) dV, where f(x,y,z) is the function representing the solid and dV is the infinitesimal volume element.

2. How is the volume of a solid represented graphically in multivariable calculus?

The volume of a solid can be represented graphically in multivariable calculus using a three-dimensional coordinate system. The solid is then represented as a region bounded by a surface in the three-dimensional space. The volume of the solid is equal to the integral of the function representing the solid over this region.

3. What is the significance of the volume of a solid in multivariable calculus?

The volume of a solid in multivariable calculus is an important concept in mathematics and physics. It is used to calculate the amount of space occupied by a three-dimensional object and is essential in solving problems related to fluid dynamics, electromagnetism, and other areas of science and engineering.

4. What factors can affect the volume of a solid in multivariable calculus?

The volume of a solid in multivariable calculus can be affected by various factors, such as the shape and size of the solid, the function representing the solid, and the boundaries of the region in which the solid is located. Additionally, changes in the values of the variables used in the function can also impact the volume of the solid.

5. How is the volume of a solid calculated for irregularly shaped objects in multivariable calculus?

The volume of a solid for irregularly shaped objects in multivariable calculus can be calculated using techniques such as slicing and shell methods, which involve breaking down the solid into smaller and simpler shapes that can be integrated over to find the total volume. These methods are often used when the function representing the solid is difficult to integrate using the triple integral formula.

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