What is the area under the curves of y=3, 2y=5(x)^1/2 and 2y+x=6

- anonymous

What is the area under the curves of y=3, 2y=5(x)^1/2 and 2y+x=6

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- anonymous

is there a picture attached?

- anonymous

Ah well it's just that graphing this is going to be a pain.. So let's see what we can do

- anonymous

no but I can attach one

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- anonymous

if u can, please do so haha

- anonymous

k here it is

##### 1 Attachment

- anonymous

omg lol more of these webworks... i'm so sick of them haha

- anonymous

sorry

- anonymous

nah it's fine.. actually give the wolframalpha link, they also list the intercepts. it's easier.

- anonymous

http://www.wolframalpha.com/input/?i=Graph+y%3D3%2C+2y%3D5x^%281%2F2%29%2C+2y%2Bx%3D6

- anonymous

all right great thanks. Here we go..

- anonymous

So here I'm going to split the graph in two at the point where x+2y = 6 and 2y = 5sqrt(x) intersect

- anonymous

okay we did that

- anonymous

That's the upper limit of our first integral

- anonymous

okay

- anonymous

Sorry my browser's having trouble, anyway the upper intersect was 1. Now the lower intersect, which occurs when y =3 and 2y+x=6 intersect.

- anonymous

6+x=6, x = 0. So the first half of that area is:
Integral (upper curve - lower curve, 0,1)

- anonymous

how did you get one I got 1.36 doesnt it turn into a quadratic

- anonymous

x+5sqrt(x) = 6? Well I mean take a look, what number plus 5 times it's square root is 6?

- anonymous

okay i get it i never thought of it like that

- anonymous

Keep in mind, we're still calculating the first half. So now, the upper curve minus the lower curve (well they're lines, but whatever) is: 3 - (6-x)/2

- anonymous

So so far, the first half of our area is:
Integral (3 - (6-x)/2, 0,1)

- anonymous

Now let's find the 2nd half:
This time the lower limit is 1, because we are starting at one, and the upper limit is the intersection of y=3 and y = 5/2sqrt(x)

- anonymous

(5/2)*sqrt(x) = 3
sqrt(x) = 6/5 => x = 36/25

- anonymous

So 2nd integral is again, upper curve minus the lower curve from 1 to 36/25, or
Integral (3 - (5/2)*sqrt(x), 1, 36/25)

- anonymous

Calculate each integral and then add them to get 107/300 (lol I cheated, used wolf to compute)

- anonymous

so it is .356666667

- anonymous

Yea that looks right.

- anonymous

k i will try it and tell you if it works

- anonymous

it works thanks sooooo much you have no idea

- anonymous

np.

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