Conceptual double integral question

In summary, the conversation discusses different approaches to solving an integral with a given region and how the same answer was obtained despite using different bounds for the inner integral. It is unclear if the book's solution included the negative region.
  • #1
Mdhiggenz
327
1

Homework Statement



∫∫x2sin(y2)dA; R is the region that is bounded by y=x3
y=-x3, and y=8.

While working out the regions for this integral I set the inner integral to -y1/3 to y1/3. and the outer integral from 0 to 8. The book however set the inner integral from 0 to y1/3. However we both got the same answer, is there a reason for this?

Thanks

Higgenz


Homework Equations





The Attempt at a Solution

 
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  • #2
Hi Mdhiggenz! :smile:
Mdhiggenz said:
While working out the regions for this integral I set the inner integral to -y1/3 to y1/3. and the outer integral from 0 to 8. The book however set the inner integral from 0 to y1/3. However we both got the same answer, is there a reason for this?

Either one of you made a mistake, or the book multiplied by 2 after integrating. :confused:
 
  • #3
Hey Tim,

Yea the books answer is quite strange, the way I reasoned my answer is when I drew the graph R2: goes from -y^1/3 to positive y^1/3. Unless the book chose not to include the negative region.

Thoughts?
 
  • #4
Your method looks fine. :confused:
 

1. What is a conceptual double integral?

A conceptual double integral is a mathematical concept used in calculus to calculate the volume under a two-dimensional surface. It involves integrating a function of two variables over a region in a coordinate plane.

2. How is a double integral different from a single integral?

A single integral calculates the area under a curve on a one-dimensional axis, while a double integral calculates the volume under a surface on a two-dimensional plane.

3. What is the purpose of using a double integral?

The purpose of using a double integral is to find the volume of a three-dimensional object, such as a solid shape, by integrating a function over a two-dimensional region.

4. What are the steps involved in solving a conceptual double integral?

The first step is to determine the limits of integration for both variables. Then, the integral is set up by multiplying the function by the infinitesimal area element, which is typically written as dA or dx dy. Finally, the integral is evaluated using the fundamental theorem of calculus.

5. Can a double integral be used to calculate other quantities besides volume?

Yes, a double integral can also be used to calculate other quantities such as surface area, mass, and moments of inertia. It can also be used in physics and engineering to solve problems involving two-dimensional systems.

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