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Thread: The area of plane figure

  1. #1
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    Exclamation The area of plane figure

    1. The plane curve is defined with a polar equation $\displaystyle r = \cos(\phi)(4\sin^2(\phi) - 1)$. Find the area of this plane figure, bound by the curve.

    2. We have the region $\displaystyle \mathbb{R}^3$ given by inequalities $\displaystyle x + y + z \leq 2, x \geq 0, y \geq 0, z \geq 0$. Find the volume of this region with the cut of slices in appropriate direction.

    3. The curve $\displaystyle \mathcal{K}$ is given parametrically by $\displaystyle x = e^{-t}\cos(t), y = e^{-t}\sin(t)$.
    a) Find the arc length of $\displaystyle \mathcal{K}$ on interval $\displaystyle t \in [0, \ln(2)]$.
    b) Find the area of the region, described by a vector from $\displaystyle A(0,0,0)$ up to the curve $\displaystyle \mathcal{K}$ on the interval $\displaystyle t \in [0, \ln(2)]$.
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    Re: The area of plane figure

    1.

    there is one loop in the first quadrant

    its area = $\displaystyle \frac{1}{2}\int_{\pi /6}^{\pi /2} r^2 \, d\phi $
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  3. #3
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    Re: The area of plane figure

    How did you come to bounds of $\displaystyle \dfrac{\pi}{6}$ and $\displaystyle \dfrac{\pi}{2}$? What about other exercises?
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    Re: The area of plane figure

    Quote Originally Posted by lebdim View Post
    1. The plane curve is defined with a polar equation $\displaystyle r = \cos(\phi)(4\sin^2(\phi) - 1)$. Find the area of this plane figure, bound by the curve.

    2. We have the region $\displaystyle \mathbb{R}^3$ given by inequalities $\displaystyle x + y + z \leq 2, x \geq 0, y \geq 0, z \geq 0$. Find the volume of this region with the cut of slices in appropriate direction.

    3. The curve $\displaystyle \mathcal{K}$ is given parametrically by $\displaystyle x = e^{-t}\cos(t), y = e^{-t}\sin(t)$.
    a) Find the arc length of $\displaystyle \mathcal{K}$ on interval $\displaystyle t \in [0, \ln(2)]$.
    b) Find the area of the region, described by a vector from $\displaystyle A(0,0,0)$ up to the curve $\displaystyle \mathcal{K}$ on the interval $\displaystyle t \in [0, \ln(2)]$.
    What work have you done on any of these? This is a math help forum, so you
    should show what you know on a problem so that someone may help you. If you cannot do any of the work, then we are not to do the work for you. You would need
    to see your instructor whether to stay with this class and/or get a tutor in person.
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    Re: The area of plane figure

    For the 2. exercise I don't know how to start ...

    3. a) $\displaystyle \int_{0}^{\ln(2)}e^{-t}dt$?

    b) How to setup the integral here?

    @greg1313
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    Re: The area of plane figure

    Quote Originally Posted by lebdim View Post

    b) How to setup the integral here?

    @greg1313
    I'll refer that to someone else if they decide to reply to it.
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    Re: The area of plane figure

    Quote Originally Posted by lebdim View Post
    For the 2. exercise I don't know how to start ...

    3. a) $\displaystyle \int_{0}^{\ln(2)}e^{-t}dt$?

    b) How to setup the integral here?

    @greg1313
    3
    b)

    integrate y(t)x'(t), t from ln(2) to 0
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    Re: The area of plane figure

    Quote Originally Posted by lebdim View Post
    For the 2. exercise I don't know how to start ...

    3. a) $\displaystyle \int_{0}^{\ln(2)}e^{-t}dt$?

    b) How to setup the integral here?

    @greg1313
    Do an obvious u-substitution and show us what you get.
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