How Do You Integrate y^3 Over a Triangle with Given Vertices?

In summary: So it's:\begin{aligned}\iint_D y^3dA &= \int_1^2 \int_{\text{weird range of length $(8 - 4y)$}} y^3dx dy \\&= \int_1^2 y^3 \left(\int_{\text{weird range of length $(8 - 4y)$}} dx\right) dy \\&=\int_1^2 y^3 (8-4y)dy\end{aligned}Yep. (Nod)In summary, the equation of AB for $y=1$ is y-1=\
  • #1
ineedhelpnow
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$\int \, \int_{D}^{} \, y^3 dA$ D is the triangular region with vertices (0,1), (1,2), (4,1)

i can't get past this problem. i drew the triangle but i don't know how to find the intervals...
excuse my ugly drawing :p
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  • #2
ineedhelpnow said:
$\int \, \int_{D}^{} \, y^3 dA$ D is the triangular region with vertices (0,1), (1,2), (4,1)

i can't get past this problem. i drew the triangle but i don't know how to find the intervals...
excuse my ugly drawing :p
View attachment 3575

Hi! ;)

Let's start at $y=1$.
We can draw a thin horizontal rectangle there with length $4$ and height $dy$.

At $y=2$, we would have a rectangle with length $0$ and height $dy$.

Everywhere between, we have a linear transition.
That is, we have a rectangle of length $(8y-4)$ and height $dy$.

That totals to:
$$\iint_D y^3 dA = \int_1^2 y^3 \cdot (8y-4) \cdot dy$$
 
  • #3
hey ils.

uuum (Blush) its a double integral though.
 
  • #4
ineedhelpnow said:
hey ils.

uuum (Blush) its a double integral though.

Yep. (Nod)

So it's:
\begin{aligned}\iint_D y^3dA &= \int_1^2 \int_{\text{weird range of length $(8 - 4y)$}} y^3dx dy \\
&= \int_1^2 y^3 \left(\int_{\text{weird range of length $(8 - 4y)$}} dx\right) dy \\
&=\int_1^2 y^3 (8-4y)dy\end{aligned}
(Wasntme)
 
  • #5
(Rofl) how do i find the weird range? so like $D={(x,y)|?\le x \le ?, ? \le y \le ?}$. i was looking at an example on chegg and they used the equation from side AB and BC for the interval of x. i don't remember how to get the equation of the sides of a triangle (Blush)
 
  • #6
ineedhelpnow said:
(Rofl) how do i find the weird range? so like $D={(x,y)|?\le x \le ?, ? \le y \le ?}$. i was looking at an example on chegg and they used the equation from side AB and BC for the interval. i don't remember how to get the equation of the sides of a triangle (Blush)

What is the range of $x$ when $y=1$?
What if $y=2$?
And what if $y$ has some value in between those 2? (Wondering)We can also do it with the equations of AB and BC, but that is more complicated. (Doh)Anyway, for the equation of AB, we have the 2 points (0,1) and (1,2).
The equation for AB is given by:
$$y-y_A=\frac{y_B-y_A}{x_B-x_A}(x-x_A)$$
That is:
$$y-1=\frac{2-1}{1-0}(x-0) \quad\Rightarrow\quad y=x+1$$
(Wasntme)
 
  • #7
thank you ils :eek:
yeah the reason why i asked for the equation of the line is because this lesson is about double integrals where one of the integrals is between a range of two equations and the other is between two actual numbers.
 
  • #8
Another formulation would be:

\(\displaystyle V=\iint\limits_{R}y^3\,dA=\int_1^2 y^3\int_{y-1}^{7-3y}\,dx\,dy=\int_1^2 y^3(8-4y)\,dy\)

Just as suggested above. :D
 
  • #9
i found BC and i got $x=3y-5$.
so $ \int_{y-1}^{3y-5} \ $?
 
  • #10
ineedhelpnow said:
i found BC and i got $x=3y-5$.
so $ \int_{y-1}^{3y-5} \ $?

That doesn't look right. (Worried)
How did you find it? (Wondering)
 
  • #11
i have no idea. i did it again. its x=7-3y
 

FAQ: How Do You Integrate y^3 Over a Triangle with Given Vertices?

What is the definition of integrating y^3 over Triangle D?

Integrating y^3 over Triangle D refers to finding the volume of a three-dimensional shape formed by a triangular base and a curved surface, where the height of the shape at any given point on the base is equal to y^3.

How is the integral of y^3 over Triangle D calculated?

To calculate the integral of y^3 over Triangle D, we use the formula V = ∫∫∫ y^3 dV, where dV represents an infinitesimal volume element. This can be evaluated using triple integration over the limits of the triangular base and the curve defining the shape.

What is the significance of integrating y^3 over Triangle D in scientific research?

Integrating y^3 over Triangle D can be useful in various scientific fields, such as physics and engineering, where calculating volumes of complex shapes is necessary. It can also be used in mathematical modeling and simulations to understand and analyze real-world phenomena.

What are some real-life applications of integrating y^3 over Triangle D?

Integrating y^3 over Triangle D has numerous practical applications, such as calculating the volume of a water tank with a triangular base, determining the displacement of a curved object, and finding the mass of a 3D-printed structure with a triangular cross-section.

Are there any limitations to integrating y^3 over Triangle D?

While integrating y^3 over Triangle D can be a useful tool, it may not always be applicable to every shape or volume. It is essential to carefully consider the limits and boundaries of the integral to ensure accurate results. Additionally, the process can be time-consuming and complex, requiring advanced mathematical knowledge and computational resources.

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