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Thread: find the area between the given curve and x axis cos (x) pi≤ x ≤ pi

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    find the area between the given curve and x axis cos (x) pi≤ x ≤ pi

    I have antiderived and gotten sin pi - sin pi = 0
    which is the correct answer ...on a graph though as it states x the area between pi and pi would that simply be a point as i imagine x between pi an pi to be so....ie the x value is not between - pi and + pi ...hmmm any help explaining would be appreciated...
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    Re: find the area between the given curve and x axis cos (x) pi≤ x ≤ pi

    Quote Originally Posted by bee77 View Post
    I have antiderived and gotten sin pi - sin pi = 0
    which is the correct answer ...on a graph though as it states x the area between pi and pi would that simply be a point as i imagine x between pi an pi to be so....ie the x value is not between - pi and + pi ...hmmm any help explaining would be appreciated...
    It is painfully clear that there is a typo or a miss-statement in the question. It should be $-\pi\le x\le\pi$
    Look at the graph. You know that the cosine function is even. Therefore the integral has to equal zero.

    Now what is the question?
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    Re: find the area between the given curve and x axis cos (x) pi≤ x ≤ pi

    find the area between the given curve and x axis cos (x) pi≤ x ≤ pi
    I assume you meant $-\pi \le x \le \pi$ (proof your posts!)

    Also, in future ... post the problem statement in the body of the post, not in the title.


    Taking advantage of the symmetry of the graph w/respect to the y-axis ...

    $\displaystyle A = \int_{-\pi}^{\pi} |\cos{x}| \, dx = 2 \bigg[\int_0^{\pi/2} \cos{x} \, dx - \int_{\pi/2}^{\pi} \cos{x} \, dx \bigg] = 4$



    btw, learn it now ... the area of one "hump" of $y=\sin{x}$ or $y = \cos{x}$ from the curve to the x-axis is 2. Check it yourself.
    Attached Thumbnails Attached Thumbnails find the area between the given curve and x axis  cos (x)  pi≤ x ≤ pi-cosx_area.jpg  
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    Re: find the area between the given curve and x axis cos (x) pi≤ x ≤ pi

    I am very greatful for all the help .Yes I am probably frustrating at times but your help will get me over the line for my exam coming up and I really appreciate the help .From the land downunder Australia cheers ...btw I will probably post more annoying help questions but you guys explain it better than my lecturer
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    Re: find the area between the given curve and x axis cos (x) pi≤ x ≤ pi

    Do you understand that the "area between" a given curve and the x-axis is not always the integral?
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    Re: find the area between the given curve and x axis cos (x) pi≤ x ≤ pi

    No not really ...what do you mean ..like a triangle or rectangle etc please explain ...cheers
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    Re: find the area between the given curve and x axis cos (x) pi≤ x ≤ pi

    Quote Originally Posted by bee77 View Post
    No not really ...what do you mean ..like a triangle or rectangle etc please explain ...cheers
    Example: $f(x) = -1$ between 0 and 1. This is a square. The area between this curve and the x-axis is obviously 1. But, $\displaystyle \int_0^1 (-1dx) = \left.-x\right|_0^1 = -1$. An integral can be positive or negative. The area between the curve and the x-axis is always positive. For any function $f(x)$, we can define two functions:

    $f^+(x) = \begin{cases}f(x) & f(x)\ge 0 \\ 0 & \text{otherwise}\end{cases}$
    $f^-(x) = \begin{cases}f(x) & f(x) \le 0 \\ 0 & \text{otherwise}\end{cases}$

    Now, the area between $f(x)$ and the x-axis on the interval $a\le x \le b$ is given by $\displaystyle \int_a^b(f^+(x)-f^-(x))dx = \int_a^b\left|f(x)\right|dx$
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