Double Integration without Anti-Derivatives

[CallMeShady]

Homework Statement

Make a good sketch of the plane region D defined by the following simultaneous inequalities:
D: y >/= -2x, 2y >/= x, 2y </= 4-x.
Use deep conceptual understanding the insight (and no antiderviative calculations!) to reduce the iterated integral below to a simple algebraic expression depending on the parameters a, b, and c:
I = ∫∫(a+bx+cy)dA.
Enter the value of I corresponding to a = -0.8, b = 0.7, and c = -1.2.

a) -7.04
b) -6.00
c) -4.67
d) 8.07
e) 2.74

Homework Equations

None, other than the integral equation provided in the question.

The Attempt at a Solution

As asked in the question, I made a quick sketch, shown below:

Then I used the dot product to figure out that the region in consideration is a right triangle. Hence, it was easy to calculate the area of this region in the 2D plane. I calculated it to be:
A = 3.097097656
Now, I don't know how to extend this idea to calculate the volume since f(x,y) is not constant.


Any help would be greatly appreciated. Thanks!

 

[haruspex]

Since two of the lines are at right angles, it would be natural to do a change of co-ordinates. This will effectively turn b and c into two different coefficients, i.e. something like ∫∫(a+b'u+c'v)dA. Can you then use a geometric interpretation of ∫∫b'u.dA etc?

 

[CallMeShady]

Since two of the lines are at right angles, it would be natural to do a change of co-ordinates. This will effectively turn b and c into two different coefficients, i.e. something like ∫∫(a+b'u+c'v)dA. Can you then use a geometric interpretation of ∫∫b'u.dA etc?

Unfortunately, I don't quite understand the idea that you are suggesting. I am unsure on how switching two co-ordinates will achieve anything in regards to this problem.

 

[SammyS]

Homework Statement

Make a good sketch of the plane region D defined by the following simultaneous inequalities:
D: y >/= -2x, 2y >/= x, 2y </= 4-x.
Use deep conceptual understanding the insight (and no antiderviative calculations!) to reduce the iterated integral below to a simple algebraic expression depending on the parameters a, b, and c:
I = ∫∫(a+bx+cy)dA.
Enter the value of I corresponding to a = -0.8, b = 0.7, and c = -1.2.

a) -7.04
b) -6.00
c) -4.67
d) 8.07
e) 2.74

Homework Equations

None, other than the integral equation provided in the question.

The Attempt at a Solution

As asked in the question, I made a quick sketch, shown below:
[ IMG]http://i61.tinypic.com/30ucktg.png[/PLAIN]
Then I used the dot product to figure out that the region in consideration is a right triangle. Hence, it was easy to calculate the area of this region in the 2D plane. I calculated it to be:
A = 3.097097656
Now, I don't know how to extend this idea to calculate the volume since f(x,y) is not constant.

Any help would be greatly appreciated. Thanks!

What does the equation, z = a+bx+cy represent ?

 

[CallMeShady]

What does the equation, z = a+bx+cy represent ?

Hm... it represents a plane. Are you suggesting that I find the volume of the two pieces (triangular base + another triangular piece on top with the slanted plane)?

 

[haruspex]

Hm... it represents a plane. Are you suggesting that I find the volume of the two pieces (triangular base + another triangular piece on top with the slanted plane)?

No, SammyS and I are both suggesting you think of it as a volume integral.

 

[LCKurtz]

I don't think your area A is correct. I suggest re-calculating it with exact values instead of decimals. Also, it isn't difficult to find the centroid of a right triangle. Your problem can be thought of as the volume of a triangular prism with slanted top.

Are you sure the values for a,b,c and the answer choices are all typed correctly?

[Edit] Never mind about the typing, all is correct except your area is only sort of close.

 

[CallMeShady]

Are you sure the values for a,b,c and the answer choices are all typed correctly?

Yes, the values are correct (as shown in my question). Althought I must point out that two previous questions have had incorrect sets of answers in this particular problem set. My professor acknowlegded them as well.

 

[LCKurtz]

Yes, I had a mistake in calculations I found. Still, you should use exact values, certainly for the area of the triangle. And the answers do have a correct choice.

 

[CallMeShady]

Well, I am still a bit confused as to where to go from here. After reading the responses and doing some research online, I stumbled upon something called the "Jacobian." Am I drifting off on a tangent to infinity?

 

[Ray Vickson]

Homework Statement

Make a good sketch of the plane region D defined by the following simultaneous inequalities:
D: y >/= -2x, 2y >/= x, 2y </= 4-x.
Use deep conceptual understanding the insight (and no antiderviative calculations!) to reduce the iterated integral below to a simple algebraic expression depending on the parameters a, b, and c:
I = ∫∫(a+bx+cy)dA.
Enter the value of I corresponding to a = -0.8, b = 0.7, and c = -1.2.

a) -7.04
b) -6.00
c) -4.67
d) 8.07
e) 2.74

Homework Equations

None, other than the integral equation provided in the question.

The Attempt at a Solution

As asked in the question, I made a quick sketch, shown below:

Then I used the dot product to figure out that the region in consideration is a right triangle. Hence, it was easy to calculate the area of this region in the 2D plane. I calculated it to be:
A = 3.097097656
Now, I don't know how to extend this idea to calculate the volume since f(x,y) is not constant.


Any help would be greatly appreciated. Thanks!

For each $(x,y) \in D$, the integrand $f = a + bx + cy$ is $f = a + \vec{u} \cdot \vec{r}$, where $\vec{u} = (c,d)$, $\vec{r} = (x,y)$ and $\cdot$ is the inner product. Since integration is a 'linear' operation, we can pull the $\vec{u} \cdot (\cdots)$ outside the integral, to get
$$\text{integral } = a \int_D\, dA + \vec{u} \cdot \int_D \vec{r} \, dA.$$
Do you recognize both of these integrals, perhaps from a physics course or somewhere similar?

 

[CallMeShady]

For each $(x,y) \in D$, the integrand $f = a + bx + cy$ is $f = a + \vec{u} \cdot \vec{r}$, where $\vec{u} = (c,d)$, $\vec{r} = (x,y)$ and $\cdot$ is the inner product. Since integration is a 'linear' operation, we can pull the $\vec{u} \cdot (\cdots)$ outside the integral, to get
$$\text{integral } = a \int_D\, dA + \vec{u} \cdot \int_D \vec{r} \, dA.$$
Do you recognize both of these integrals, perhaps from a physics course or somewhere similar?

No, I don't recognize this from any of my courses. I was able to follow through the logic that you presented. However, shouldn't it be $\vec{u} = (b,c)$? Also, when you break the integral into two in the end, don't I still have to use anti-derivatives for the second piece in x and y (as contained in the vector r)?

 

[haruspex]

don't I still have to use anti-derivatives for the second piece in x and y (as contained in the vector r)?

Not if you recognize the physical interpretation of the integral and so can apply a well-known formula.
SammyS and I were pushing you towards thinking of it as a volume integral. LCKurtz also refers to this, specifically as a triangular prism. My rotation of co-ordinates suggestion was to turn it from a triangular prism into something a bit simpler. But the best suggestion is LCKurtz's mention of centroids. (Ray V might also have that in mind.)
How do you calculate the location of the centroid of a lamina?

 

[Ray Vickson]

No, I don't recognize this from any of my courses. I was able to follow through the logic that you presented. However, shouldn't it be $\vec{u} = (b,c)$? Also, when you break the integral into two in the end, don't I still have to use anti-derivatives for the second piece in x and y (as contained in the vector r)?

Yes, it should be $\vec{u} = (b,c)$. The first term should be easy; the second term will be familiar to you if you have taken a Mechanics course; otherwise, perhaps not.

 

[CallMeShady]

How do you calculate the location of the centroid of a lamina?

Well, for a right triangle, it's simple as the formulas are already dervided online. Or, we could simply integrate using (assuming density is constant):

x_centroid = 1/A * ∫∫x*dA
y_centroid = 1/A * ∫∫y*dA

Suppose I do find the centroid co-ordinates, how does that result into determining a volume? I feel like I am missing something big here but I just can't see it.

 

[LCKurtz]

Well, for a right triangle, it's simple as the formulas are already dervided online. Or, we could simple integrate using (assuming density is constant):

x_centroid = 1/A * ∫∫x*dA
y_centroid = 1/A * ∫∫y*dA

Suppose I do find the centroid co-ordinates, how does that result into determining a volume? I feel like I am missing something big here but I just can't see it.

So you could rewrite those as$$
\bar x A =\iint x~dA,~~\bar y A = \iint y~dA$$Can you use that to rewrite your problem$$
I = \iint a + bx + cy~dA~\text{?}$$

 

[SammyS]

Homework Statement

...

Use deep conceptual understanding the insight (and no antiderviative calculations!) ...

Any help would be greatly appreciated. Thanks!

...

Also, it isn't difficult to find the centroid of a right triangle. Your problem can be thought of as the volume of a triangular prism with slanted top.
...

Well, I am still a bit confused as to where to go from here. After reading the responses and doing some research online, I stumbled upon something called the "Jacobian." Am I drifting off on a tangent to infinity?

In the original statement of the problem, it's clear that you are not to evaluate the integral by using anti-derivatives.

I agree with LC Kurtz that you can calculate the volume of a triangular prism -- or maybe two prisms. (I haven't done the calculations.)

 

[LCKurtz]

In the original statement of the problem, it's clear that you are not to evaluate the integral by using anti-derivatives.

I agree with LC Kurtz that you can calculate the volume of a triangular prism -- or maybe two prisms. (I haven't done the calculations.)

You don't need to do that.

 

[SammyS]

You don't need to do that.

Oh, right!

Using the centroid takes care of all that complication !

 

[haruspex]

Suppose I do find the centroid co-ordinates, how does that result into determining a volume?

No, I meant that there are two physical interpretations available for your integral: as the volume of a prism (or tetrahedron if you do the co-ordinate rotation); or as the centroid of a lamina. The second is definitely easier.

 

[CallMeShady]

Interesting stuff.

So I broke up the integral and recognized this:
I = a∫∫dA + b∫∫xdA + c∫∫ydA.

The first terms is simple "a" times the area, the second term is the centroid of x times the area times b, and the last terms is the centroid of y times the area times c. My centroid points appear to be correct (intuitively from the graph), but my final answer was -1.967611565. None of the options have this answer included. So, I am going to do what LCKurtz said about re-calculating the area and I'll update.

If my logic above seems incorrect, please do stop me.

 

[LCKurtz]

Interesting stuff.

So I broke up the integral and recognized this:
I = a∫∫dA + b∫∫xdA + c∫∫ydA.

The first terms is simple "a" times the area, the second term is the centroid of x times the area times b, and the last terms is the centroid of y times the area times c. My centroid points appear to be correct (intuitively from the graph), but my final answer was -1.967611565. None of the options have this answer included. So, I am going to do what LCKurtz said about re-calculating the area and I'll update.

If my logic above seems incorrect, please do stop me.

That's the idea. You should give exact values for the area and the x and y centroid values, so your formula for I is exact.

 

[CallMeShady]

That's the idea. You should give exact values for the area and the x and y centroid values, so your formula for I is exact.

Ahhhhh this is going to make me have nightmares tonight! So I found a calculation error in determining the height of the right triangle, so I re-did all the calculations with the correct values. I got an answer of -2.29. It's not one of the answers. Then I rounded my values of base, height, and the area, and re-did all calculations. This time, my answer was -2.36. Still not any of the answers.

I have no idea what I am doing wrong.

 

[LCKurtz]

Ahhhhh this is going to make me have nightmares tonight! So I found a calculation error in determining the height of the right triangle, so I re-did all the calculations with the correct values. I got an answer of -2.29. It's not one of the answers. Then I rounded my values of base, height, and the area, and re-did all calculations. This time, my answer was -2.36. Still not any of the answers.

I have no idea what I am doing wrong.

At this point there is nothing more we can do to help you until you post your work instead of just giving answers. Show us your calculations for the two sides of the triangle and its centroid. Also show us your exact formula for I in terms of (a,b,c).

 

[CallMeShady]

http://i62.tinypic.com/zxtvsx.jpg


Might be hard to read, but maybe it may be visible enough? Please click on the image, and then select the option (view raw image) to zoom in.

 

[SammyS]

Do you have the coordinates for the centroid of the base?

 

[haruspex]

The equations x' = x cos(θ) + y sin(θ), y' = -x sin(θ) + y cos(θ), are to rotate a point about the origin, right? I would have guessed you'd be (notionally) rotating the triangle clockwise to align with the axes, finding the centroid of that, then rotating back anticlockwise. But (x, y) is not that centroid (a factor 3 too much, so actually the far corner of the rectangle of which the aligned triangle is one half?), and the rotation you've used is clockwise, not anticlockwise.

 

[CallMeShady]

Do you have the coordinates for the centroid of the base?

The centroid co-ordinates are (1.110899311, 0.555130063). The first co-ordinate is for the base. Basically what I did was position the right triangle's base on the x-axis, and then find the centroid co-ordinates of it. Then I rotated the triangle by 26.5 degrees counter clockwise in order to obtain the actual centroid co-ordinates to match the question.

 

[SammyS]

(1.110899311, 0.555130063) are not the centroid coordinates for the figure in your original post .

 

[CallMeShady]

And as haruspex mentioned, the formula I was using did not rotate the co-ordinate counterclockwise. A quick Google search led me to the correct formula, and now I can satisfyingly say that I found the correct answer:

I = -7.027811891

It's off by 0.01 from one of the answers, but that should be close enough. Assuming that this is, indeed, the correct answer, I thank you all for your help. You guys are awesomely patient and helpful. :D

 

[LCKurtz]

Well I, for one, would not be satisfied with what you have done, and you shouldn't be either, even though you found the correct answer. Once you have the exact coordinates for the three vertices of the right triangle, it is an easy matter to get the exact coordinates of the centroid since it is 1/3 of each leg in the direction of each leg. You don't need trig functions or to rotate anything. And it is also easy to calculate the area exactly and to write down an exact formula for your integral. And the answers just involve simple fractions. You would only get partial credit from me.

 

[SammyS]

Well I, for one, would not be satisfied with what you have done, and you shouldn't be either, even though you found the correct answer. Once you have the exact coordinates for the three vertices of the right triangle, it is an easy matter to get the exact coordinates of the centroid since it is 1/3 of each leg in the direction of each leg. You don't need trig functions or to rotate anything. And it is also easy to calculate the area exactly and to write down an exact formula for your integral. And the answers just involve simple fractions. You would only get partial credit from me.

Yes, LC makes an excellent point here.

According to the Original Post, the instructions for doing this problem included the following.

"Use deep conceptual understanding the insight (and no antiderviative calculations!) to reduce the iterated integral below to a simple algebraic expression depending on the parameters a, b, and c:"

(emphasis added by me)​

 

[CallMeShady]

Well I, for one, would not be satisfied with what you have done, and you shouldn't be either, even though you found the correct answer. Once you have the exact coordinates for the three vertices of the right triangle, it is an easy matter to get the exact coordinates of the centroid since it is 1/3 of each leg in the direction of each leg. You don't need trig functions or to rotate anything. And it is also easy to calculate the area exactly and to write down an exact formula for your integral. And the answers just involve simple fractions. You would only get partial credit from me.

I can calculate the area exactly without rotation; however, I could not think of a way to find the centroid of the triangle in the given orientation in the question. Hence, by making a simple rotation, it just made my life easier.

 

[LCKurtz]

Once you have the exact coordinates for the three vertices of the right triangle, it is an easy matter to get the exact coordinates of the centroid since it is 1/3 of each leg in the direction of each leg.

I can calculate the area exactly without rotation; however, I could not think of a way to find the centroid of the triangle in the given orientation in the question. Hence, by making a simple rotation, it just made my life easier.

Didn't I just tell you how to find it?