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find the area of the part of the circle r=9sinθ+cosθ in the fourth quadrant?

area of the part = ...........

Update:

how would be 0 for example for 2sinθ+cosθ its 0.07956 and for4sinθ+cosθ its 0.04116

4 Answers

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  • 1 year ago
    Favorite Answer

    This is pretty easy to convert to Cartesian coordinates. Multiply both sides by r to get

    r² = 9r sin θ + r cos θ

    x² + y² = 9y + x

    x² - x + y² - 9y = 0

    x² - x + (1/2)² + y² - 9y + (9/2)² = 1/4 + 81/4

    (x - 1/2)² + (y - 9/2)² = 82/4

    That's a circle centered at (1/2, 9/2) with a radius of (√82)/2, slightly larger than 9/2. Set y=0 to solve for the x-intercepts of the circle:

    (x - 1/2)² + 81/4 = 82/4

    (x - 1/2)² = 1/4

    |x - 1/2| = 1/2

    x = 0 or 1

    So, none of the area in the circle and below the x-axis is in the 3rd quadrant. Find y as a function of x from the circle equation above:

    (y - 9/2) = - √[82/4 - (x - 1/2)²] .... negative square root on right since lower arc of circle is wanted

    y = 9/2 - √[82/4 - (x - 1/2)²]

    Integrate y dx from x=0 to 1 and you have your answer.

    • munnaam1 year agoReport

      its correct thanks
      its
      0.01847303406486075

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  • 1 year ago

    Has a diameter from (0,9) to (1,0), and the circle passes through (0,0).

    Radius: √(9²+1²)/2 = √(41/2)

    Area of the sector including this segment: (41/4)arccos((2(41/2) - 1)/(2*(41/2))

    Area of the isosceles triangle in the sector including this segment: (9/4)

    Area of the segment (in Qiv) is (41/4)arccos(40/41) - 9/4 ≈ 0.018 units²

    Note: This is a very small protrusion into the 4th quadrant. Please visit the graph below, and zoom in around (0.5,0) for more detail: https://www.desmos.com/calculator/nxcpwsic9b

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  • iceman
    Lv 7
    1 year ago

    A = ∫1/2(9sinθ+cosθ)^2 dθ [3pi/2, 2pi] = (41pi )/4 - 9/2 ≈ 27.7

    I hope this helps.

  • Ian H
    Lv 7
    1 year ago

    Was this a "trick" question or am I missing the meaning?

    Here is the graph of that circle and as you can see it is entirely in the first two quadrants.

    https://www.wolframalpha.com/input/?i=r%3D9sin%CE%...

    So, the answer would be zero.

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