Calculus challenge

The type of bread chosen for this special calculus toast isn't the square sandwich shape, but the kind that is curved across the top. Imagine that the toast is composed of the curved part sitting atop the rectangular portion. The equation of the curved part of the toast is x2/4 + y2 = 1, and it sits directly and perfectly on top of a rectangle of height 3 inches. a) What are the equations of the rectangular boundaries? b) Graph the toast boundaries, making certain to include screen shots of the boundary equations, Window settings, and the graph. c) How would you find the length of the curve

ok

then

once you find two x values , they are your answer for a)

is x two negative values

is x squred?

ya here is the equation again \[\frac{ x^2 }{ 4 } + y^2 = 1\]

ahh, that's different, this is ellipse x^2/4 + (y-3)^2 = 1

now plug in y=3, x^2/4=1 x^2=4 x=2,-2 that's your part a

sweet ur r doing grt go ahead

part b is graphing,

Here's the graph, done without benefit of the above

i wud just graph that right

oh thx @dlipson1 can u go further

Uh oh, I think that's a line integral, I'd have to look that up. Do you know anything about Stochastic Optimization?

hey hey hey never mind...i know how to find the length of the curve thanks..but i dont knw do i need to find length of whole curve or just the bread as in ur graph

Well, the rectangle is trivial (is the side along the x-axis included?), the top is just half the ellipse, that's the only real calculus (integration) you have to do.

so from negative 2 to 2...right

Yeah, use the top half, y = sqrt(...), then (I just looked it up): Length = integral (sqrt(1+(y')^2))dy

ya and what kinda graphing calculator r u using dude

http://www.padowan.dk/ and Google --> http://tutorial.math.lamar.edu/Classes/CalcII/ArcLength.aspx

thx...getting my next question...i wud give u a lot of awards but unfortunately this site doesnt aloow lol

"Graph" (from padowan.dk) gives me the curve length of 4.882, then +3+3 +4 for the entire perimeter.

sweet thanks

hey @dlipson1 one more thing, how wud we find area on top of the toast

The rectangle + 2*integral (by symmetry) from 0 to 2 of y = sqrt(1-x^2/4)... hmm, do we need substitution here?

do u need the derivative...i have it

-x/(4y-12)....now wht to do

I'm not sure dy/dx helps. I set up the integral, thought about a trig. substitution, multiplied through by the 2 (as sqrt(4)) to get int(sqrt(4-x^2))dx, which I found here, but I think there must be an easier way (like polar coordinates): http://answers.yahoo.com/question/index?qid=20071231235113AAAAfuP

Here is almost the same problem: http://www.youtube.com/watch?v=PSlsj0IP8R8

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