Finding the area between y = 2sqrt(x), y = 4, and y = -2x + 4

[s3a]

Homework Statement

Problem:
Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Then find the
area of the region.

Given curves:
y = 2sqrt(x), y = 4, and y = -2x + 4

Technically, the curves are given as follows (but, I simplified them):
2y = 4sqrt(x), y = 4, 2y + 4x = 8

This ( http://www.wolframalpha.com/input/?i=find+area+between+y+=+2sqrt(x),+y+=+4,+y+=+-2x+++4 ) Wolfram Alpha link shows the plot for the area wanted.

Homework Equations

Definite integration and drawing functions.

The Attempt at a Solution

Could someone please tell me why that part that looks like a triangle which is from x = 0 to x = 4 and y = 2 to y = 4 is not shaded?

 

[SteamKing]

Because it doesn't fall within the region of integration.

You have specified three curves. The Wolfram Alpha apparently chose the region where y is >= 2*sqrt(x) and y <= 4-2x. If you want a different region, you will have to give WA a different specification on what to integrate.

BTW, please follow the HW template. Stuffing the problem statement into the thread title is not recommended.

 

[s3a]

BTW, please follow the HW template. Stuffing the problem statement into the thread title is not recommended.

Sorry, I assumed people would see the equations in the Wolfram Alpha link as they look at the plot; I added it now in the opening post.

If you want a different region, you will have to give WA a different specification on what to integrate.

I'm just trying to answer a problem that says to find the area between those three functions. In other words, I want to do what the problem intends for me to do. I thought this meant to find ONLY the part that looks like a triangle which is from x = 0 to x = 4 and y = 2 to y = 4 and is not shaded in Wolfram Alpha's plot. Could you please tell me what exactly the problem intends for me to do?

 

[s3a]

BTW, please follow the HW template. Stuffing the problem statement into the thread title is not recommended.

Sorry, I assumed people would see the equations in the Wolfram Alpha link as they look at the plot; I added it now in the opening post.

If you want a different region, you will have to give WA a different specification on what to integrate.

I'm just trying to answer a problem that says to find the area between those three functions. In other words, I want to do what the problem intends for me to do. I thought this meant to find ONLY the part that looks like a triangle which is from x = 0 to x = 4 and y = 2 to y = 4 and is not shaded in Wolfram Alpha's plot. Could you please tell me what exactly the problem intends for me to do?

 

[Ray Vickson]

Homework Statement

Problem:
Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Then find the
area of the region.

Given curves:
y = 2sqrt(x), y = 4, and y = -2x + 4

Technically, the curves are given as follows (but, I simplified them):
2y = 4sqrt(x), y = 4, 2y + 4x = 8

This ( http://www.wolframalpha.com/input/?i=find+area+between+y+=+2sqrt(x),+y+=+4,+y+=+-2x+++4 ) Wolfram Alpha link shows the plot for the area wanted.

Homework Equations

Definite integration and drawing functions.

The Attempt at a Solution

Could someone please tell me why that part that looks like a triangle which is from x = 0 to x = 4 and y = 2 to y = 4 is not shaded?

For this type of problem I strongly urge you to avoid the use of Wolfram Alpha: It is important to know how to do these things manually. Besides that, it is much easier to do it by hand than to try to finesse Alpha to do what you want. Of course, a hand-drawn sketch will be crude and approximate, but that should be enough to be getting on with.

 

[s3a]

For this type of problem I strongly urge you to avoid the use of Wolfram Alpha: It is important to know how to do these things manually. Besides that, it is much easier to do it by hand than to try to finesse Alpha to do what you want. Of course, a hand-drawn sketch will be crude and approximate, but that should be enough to be getting on with.

I DID draw it manually for the sake of knowing how to do it but, I didn't want to show you guys my piece-of-garbage drawing. ;) (In other words, it's the same thing as Wolfram Alpha's drawing, just uglier.)

So ... was I right (and Wolfram Alpha wrong) about what area I need to compute?

 

[Ray Vickson]

Sorry, I assumed people would see the equations in the Wolfram Alpha link as they look at the plot; I added it now in the opening post.


I'm just trying to answer a problem that says to find the area between those three functions. In other words, I want to do what the problem intends for me to do. I thought this meant to find ONLY the part that looks like a triangle which is from x = 0 to x = 4 and y = 2 to y = 4 and is not shaded in Wolfram Alpha's plot. Could you please tell me what exactly the problem intends for me to do?

OK, so now you need to compute the area of that almost-triangular region. You are intended to apply one of the formulas you have probably learned for doing that.

Just as a reminder, though: you can split the area up into a large number of narrow vertical rectangles or a large number of narrow horizontal rectangles. Choose whichever method you like best.

 

[s3a]

Okay so, this (check the TheAreasToCompute.jpg attachment) is how I compute the area(s) I want, right?