Question

Urgent! Would anyone know how to find the area of a parallelogram using integration. I am given the 4 vertices (2,1) (5,4) (6,1) and (9,4) and have drawn the diagram. From this I also get the two functions y=x+1 and y=x+5
Not sure what to do from here.

Answer 1

Milko: Have you tried dividing the parallelogram into horiz. strips of width dy?

Answer 2

Yes I thought about that, and I will get three areas.
That will give me three integrals and I would add these up.
So to work it out go right curve minus left curve?

Answer 3

Assuming that your two formulas for y are correct: Solve each for x:
x=y-1 and x=y-5

Answer 4

Ok I will attach a diagram in a moment

Answer 5

sorry; my connection failed, and OpenStudy was malfunctioning.
Once you've solved both relationships for x, decide which one represents the right boundary of your parallelogram and whihc the left.

Answer 6

Nice diagram!
subtranct the equation repres. the left boundary from the one repres. the right boundary.
The result is the length of each of your horiz strips. What is it?

Answer 7

Very, very happy to hear that this discussion has been helpful for you!

Answer 8

Lost my connection...yet again!

My result: the horiz strips are all 4 units long. Agreed or not?

Answer 9

Would my integral just be [x+5] - [x+1] and then I integrate this, and my limits are from 0 to 1 ?
so the once integrated it is [x^2/2+5x]-[x^2/2+x] and once the limits are subbed in it gives me 4.
4x3 strips = 12 ???

Answer 10

Since we're working with horiz. strips of width dy, your limits of integration should be y=1 to y=4, and instead of just 3 horiz strips, y ou'll be working with an infinite number of horiz. strips.

Answer 11

What we want is Int(4dy) from 1 to 4.
This boils down to 4*Int(dy) from 1 to 4.
Make any sense?

Answer 12

I get the result 12, as you do. But help me understand your reasoning: why 3 horiz stirips instead of an infinite # of horiz strips?

Answer 13

I am not sure, the question says to do it by separating into 3 regions. But the 1 to 0 I got by the graph, each region increased in the y value by 1, so it's the same thing I guess?

Answer 14

Yes, because your geometric figure is so regular. Are you attempting to do this problem using integral calculus, or some other means?

Answer 15

& I think I have to change my equations around. Essentially the answer will be the same because it's a parallelogram but the working should be correct.

Answer 16

Note that the simplest formula for the area of a parallelogram such as this one is A=bh.
Here the length b is 4 and the height h is 3, and so the area is 3*4=12, confirming tha tyour result is the correct one.

Answer 17

so it should be x=y-1 and x=y-5

yep unfortunately no marks for doing it that way. if only!

Answer 18

Yes. If you were to subtract one of those 2 expressions for x from the other, what would you get? The result (or the absolute value of the result) will be 4, which is the length of any thin horizontal strip within the geometric figure.

Answer 19

Thank you!

Answer 20

My great pleasure. 'Til later!