Calc III Double Integral Question

In summary, the problem at hand is to write a single integral and evaluate it, but the difficulty lies in setting the bounds for a compounded integral. The graph of the region is a shaded area, which appears to be a Type 2 region (dxdy). Initially, the y bounds were thought to be from 0 to √2 and the x bounds from x=sqrt(1-x^2) to x=sqrt(4-x^2). However, it was realized that these bounds included extra space outside the intended region, and the y=x line must be involved in some way. Ultimately, the solution lies in writing the integral in polar form, rather than Cartesian form. The answer for this problem is 15/
  • #1
dropoutofschool
3
0
View attachment 96628
96628.gif


This is the problem I'm trying to solve. The directions require me to rewrite as a single integral and evaluate. But I'm having trouble setting the bounds for a complete compounded integral. The graph of the region would look something like this...
View attachment 96629
96629.png

Where the shaded area is the region. I would think its a Type 2 region (dxdy). The y bounds would be 0 to√2 and the x bounds would be from x=sqrt(1-x^2) to x=sqrt(4-x^2)... so so I thought. I then realized that these bounds would extra space outside of the intended shaded region, so the y=x linemust be involved in the bounds somehow.

I just need help setting the integral up, the actual integrating should be easy. Thanks ! Any help is appreciated!
 
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  • #2
Ugh Pictures didn't post... please refer to this...
Hw.GIF


pic.png
 
  • #3
dropoutofschool said:
View attachment 96628
This is the problem I'm trying to solve. The directions require me to rewrite as a single integral and evaluate. But I'm having trouble setting the bounds for a complete compounded integral. The graph of the region would look something like this...
View attachment 96629
Where the shaded area is the region. I would think its a Type 2 region (dxdy). The y bounds would be 0 to√2 and the x bounds would be from x=sqrt(1-x^2) to x=sqrt(4-x^2)... so so I thought. I then realized that these bounds would extra space outside of the intended shaded region, so the y=x linemust be involved in the bounds somehow.

I just need help setting the integral up, the actual integrating should be easy. Thanks ! Any help is appreciated!

I'm pretty sure the idea is to write the integral in polar form, not in Cartesian form... There's no way you can write this integral in Cartesian form with only one integral.
 
Last edited:
  • #4
In the future, please post questions like this in the Homework & Coursework sections (under Calculus), not here in the technical math sections.
 
  • #5
WOW! Completely forgot about polar, just completed the problem----for future reference, the answer is 15/16. Thank you sir and will change where I post next time!
 

Related to Calc III Double Integral Question

1. What is a double integral?

A double integral is a type of mathematical operation that involves finding the volume under a surface in a three-dimensional space. It is essentially the integral of an integral, and is typically used in multivariable calculus to solve problems involving multiple variables.

2. How do you solve a double integral?

To solve a double integral, you first need to determine the limits of integration for both variables. Then, you can use various techniques such as iterated integrals or change of variables to evaluate the integral. It is important to carefully set up the integrand and follow the proper steps in solving the integral.

3. What is the difference between a single and double integral?

A single integral involves finding the area under a curve in a two-dimensional space, while a double integral involves finding the volume under a surface in a three-dimensional space. In other words, a single integral has one variable while a double integral has two variables.

4. When do you use a double integral?

A double integral is used when solving problems that involve multiple variables, such as finding the mass or center of mass of a three-dimensional object, calculating work done by a force, or finding the average value of a function over a given region in space.

5. What are some common applications of double integrals?

Some common applications of double integrals include calculating volumes and surface areas of three-dimensional objects, finding the center of mass of a solid, calculating work and flux, and solving optimization problems in physics, engineering, and economics.

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