Integral on plane inside a cylinder

In summary, the surface $\Sigma$ defined by $(x,y,4+x+y)$ is inside the cylinder with equation $x^2+y^2=4$ and calculates $\iint_{\Sigma}(x^2+y^2)zdA$ which equals $\iint_{D}(x^2+y^2)(4+x+y)dxdy$.
  • #1
mathmari
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Hey! :eek:

I want to calculate $\iint_{\Sigma}(x^2+y^2)zdA$ on the part of the plane with equation $z=4+x+y$ that is inside the cylinder with equation $x^2+y^2=4$.

We can define the surface $\Sigma : D\rightarrow \mathbb{R}^3$ with $\Sigma (x,y)=(x,y,4+x+y)$, where $D$ is the space that is defined by $x^2+y^2\leq 4$, i.e. $D=\{(x,y)\mid x^2+y^2\leq 4\}$, right?

So, we get the following:
$$\iint_{\Sigma}(x^2+y^2)zdA=\iint_{D}(x^2+y^2)(4+x+y)dxdy$$

How can we continue? Do we use here cylindrical coordinates? (Wondering)
 
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  • #2
Using "dxdy" as the differential you are projecting down to the xy-plane. That is the circle [tex]x^2+ y^2= 4[/tex] which you can cover by taking x from -2 to 2 and, for each x, y from [tex]-\sqrt{4- x^2}[/tex] to [tex]-\sqrt{4- x^2}[/tex]:
[tex]\int_{-2}^2\int_{-\sqrt{4- x^2}}^{\sqrt{4- x^2}} (x^2+ y^2)(x+ y+ 4) dydx[/tex]

Or, in polar coordinates:
[tex]\int_0^{2\pi}\int_0^2 r^2(rcos(\theta)+ rsin(\theta)+ 4)r drd\theta= \left(\int_0^{2\pi} cos(\theta)d\theta\right)\left(\int_0^2 r^3 dr\right)+ \left(\int_0^{2\pi} sin(\theta)d\theta\right)\left(\int_0^2 r^3dr\right)+ 8\pi\int_0^2 r^3 dr= 8\pi\int_0^2 r^3 dr[/tex]
 
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  • #3
mathmari said:
Hey! :eek:

I want to calculate $\iint_{\Sigma}(x^2+y^2)zdA$ on the part of the plane with equation $z=4+x+y$ that is inside the cylinder with equation $x^2+y^2=4$.

We can define the surface $\Sigma : D\rightarrow \mathbb{R}^3$ with $\Sigma (x,y)=(x,y,4+x+y)$, where $D$ is the space that is defined by $x^2+y^2\leq 4$, i.e. $D=\{(x,y)\mid x^2+y^2\leq 4\}$, right?

Yep. (Nod)

mathmari said:
So, we get the following:
$$\iint_{\Sigma}(x^2+y^2)zdA=\iint_{D}(x^2+y^2)(4+x+y)dxdy$$

What happened to $\| \Sigma_x \times \Sigma_y \|$? (Wondering)

mathmari said:
How can we continue? Do we use here cylindrical coordinates?

That is advisable yes.
As HallsofIvy showed, it's fairly straight forward in cylindrical coordinates.
Working out the cartesian expression is probably doable, but it becomes messy pretty quick. (Nerd)
 
  • #4
Thank you so much! (Yes)
 

FAQ: Integral on plane inside a cylinder

1. What is the purpose of integrating on a plane inside a cylinder?

The purpose of integrating on a plane inside a cylinder is to calculate the volume of a shape that is enclosed within a cylindrical boundary. This is useful in many engineering and scientific applications, such as calculating the volume of a liquid in a cylindrical tank or the amount of material needed to fill a cylindrical container.

2. How is the integral on a plane inside a cylinder calculated?

The integral on a plane inside a cylinder is calculated by breaking down the shape into infinitesimally small slices, calculating the area of each slice, and then summing up all the areas to find the total volume. This is done using a technique called triple integration, which involves integrating over three variables (x, y, and z) to cover the entire volume of the cylinder.

3. What is the difference between integrating on a plane inside a cylinder and integrating in three-dimensional space?

The main difference between integrating on a plane inside a cylinder and integrating in three-dimensional space is the shape of the boundary. When integrating in three-dimensional space, the boundary can be any shape, while integrating on a plane inside a cylinder has a cylindrical boundary. This difference affects the variables used in the integral and the limits of integration.

4. Can the integral on a plane inside a cylinder be used for irregularly shaped objects?

Yes, the integral on a plane inside a cylinder can be used for irregularly shaped objects as long as the object can be enclosed within a cylindrical boundary. This method is often used in computer-aided design (CAD) software to calculate the volume of complex 3D shapes.

5. What are some real-world applications of integrating on a plane inside a cylinder?

Integrating on a plane inside a cylinder has many real-world applications. Some examples include calculating the volume of liquid in a cylindrical tank, determining the amount of material needed to create a cylindrical structure, and finding the volume of irregularly shaped objects that can be approximated by a cylinder. This method is also used in fluid mechanics and heat transfer calculations.

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