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Math Help - region bounded by curve

  1. #1
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    region bounded by curve

    for the region bounded by the curves y=0 and y= +squareroot(9-x^2) (only the positive portion of that function, i think thats why there is a plus sign in front of the squareroot) find:
    1. its area
    2. M sub x
    3. M sub y
    4. X bar (x with a bar on top of it)
    5. Y bar (y with a bar on top of it)
    6. the centroid

    i have no idea how to do this problem...
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by jeph View Post
    for the region bounded by the curves y=0 and y= +squareroot(9-x^2) (only the positive portion of that function, i think thats why there is a plus sign in front of the squareroot) find:
    1. its area
    2. M sub x
    3. M sub y
    4. X bar (x with a bar on top of it)
    5. Y bar (y with a bar on top of it)
    6. the centroid

    i have no idea how to do this problem...
    1. Area:

    A= integral(x=-3:3) y dx = integral(x=-3:3) sqrt(9-x^2) dx

    There are a number of ways of doing this integral, some of which you
    should find in you calculus notes. The simplest is to observe that we are
    trying to find the area of a semi-circle of radius 3, so the area is (9/2) pi.

    2. and 3, I don't know what you want M sub x or M sub y to mean so I
    cant help you.

    4. mean of x:

    mx = integral(R) x dR

    where R is the region defined and dR is the element of area.

    mx= integral(x=-3:3){integral(y=0:sqrt(9-x^2)) x dydx}.

    But we can observe that by symmetry mean of x is zero.

    5. Similarly: mean of y:

    mx = integral(R) y dR

    where R is the region defined and dR is the element of area.

    mx= integral(x=-3:3){integral(y=0:sqrt(9-x^2)) y dydx}

    RonL
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  3. #3
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    the M subs i think are either mass or moment....
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    A= integral(x=-3:3) y dx = integral(x=-3:3) sqrt(9-x^2) dx

    do you think you can integrate that for me? i know you could just use algebra but im curious...ive had 3 different tutors try to integrate it without using the long formula behind the book and they were all unable to get 9pie/2. they keep getting 0, 6 or -6, they can't figure how you even get a pi in the answer...
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  5. #5
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by jeph View Post
    A= integral(x=-3:3) y dx = integral(x=-3:3) sqrt(9-x^2) dx

    do you think you can integrate that for me? i know you could just use algebra but im curious...ive had 3 different tutors try to integrate it without using the long formula behind the book and they were all unable to get 9pie/2. they keep getting 0, 6 or -6, they can't figure how you even get a pi in the answer...
    The simple way is to note that this integral is the upper half of a circle of radius 3, so
    Int[sqrt{9 - x^2}dx, -3, 3] = (1/2)*(pi)*3^2 = (9/2)*(pi).

    Done!

    The hard way. (Hang on to your hat!)
    Int[sqrt{9 - x^2}dx]:

    Let x = 3y ==> dx = 3dy
    Int[sqrt{9 - x^2}dx] = Int[sqrt{9 - (3y)^2}*3dy]

    = 9*Int[sqrt{1 - y^2}dy]

    Now let y = sin(t) ==> dy = cos(t) dt

    = 9*Int[sqrt{1 - sin^2(t)} * cos(t) dt]

    = 9*Int[cos^2(t) dt]

    Now, cos(2t) = 2*cos^2(t) - 1 thus cos^2(t) = (1/2)*(cos(2t) + 1)

    Thus
    Int[sqrt{9 - x^2}dx] = 9*Int[cos^2(t) dt] = (9/2)*Int[(cos(2t) + 1)dt]

    = (9/2)*Int[cos(2t)dt] + (9/2)*Int[dt]

    Let z = 2t ==> dz = 2dt
    Int[sqrt{9 - x^2}dx] = (9/2)*Int[cos(2t)dt] + (9/2)*Int[dt]

    = (9/4)*Int[cos(z)dz] + (9/2)t

    = (9/4)*sin(z) + (9/2)t

    Now back substitute:
    Int[sqrt{9 - x^2}dx] = (9/4)*sin(z) + (9/2)t

    = (9/4)*sin(2t) + (9/2)t

    = (9/4)*sin(2*sin^{-1}(y)) + (9/2)*sin^{-1}(y)

    = (9/4)*sin(2*sin^{-1}(x/3)) + (9/2)*sin^{-1}(x/3)

    Now x varies from -3 to 3:
    Int[sqrt{9 - x^2}dx, -3, 3] = [(9/4)*sin(2*sin^{-1}(3/3)) + (9/2)*sin^{-1}(3/3)] - [(9/4)*sin(2*sin^{-1}(-3/3)) + (9/2)*sin^{-1}(-3/3)]

    = [(9/4)*sin(2*sin^{-1}(1)) + (9/2)*sin^{-1}(1)] - [(9/4)*sin(2*sin^{-1}(-1)) + (9/2)*sin^{-1}(-1)]

    = [(9/4)*sin(2*(pi)/2) + (9/2)*(pi)/2] - [(9/4)*sin(2*-(pi)/2) + (9/2)*-(pi)/2]

    = [(9/4)*sin(pi) + (9/4)*(pi)] - [(9/4)*sin(-pi) + (9/4)*-(pi)]

    = [0 + (9/4)*(pi)] - [0 + (9/4)*-(pi)]

    = (9/4)*(pi) + (9/4)*(pi) = (9/2)*(pi)

    As advertised.

    -Dan
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    thats alot of work......why cant you just substitute asinx at the beginning? it looked like the formula would work. and thats what all 3 of the tutors i asked did...
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  7. #7
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by jeph View Post
    thats alot of work......why cant you just substitute asinx at the beginning? it looked like the formula would work. and thats what all 3 of the tutors i asked did...
    I don't see any reason why you couldn't. I just tend to do things one step at a time.

    -Dan
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  8. #8
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    can you do it and tell me if it works out correctly?
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