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Thread: Finding the Area Beneath Trigonometric Curves

  1. #1
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    Finding the Area Beneath Trigonometric Curves

    I know the solution to this particular question, but I'm not too sure of the reasoning behind it. The question asks;

    Two curves, $\displaystyle y = \sqrt{3}cosx$ and $\displaystyle y = sinx$, are drawn on a graph. The first two intersections to the right of the y-axis are labelled A and B.

    a) Solve the equation $\displaystyle \sqrt{3}cosx$ = $\displaystyle sinx$ to find the x-coordinates of A and B.

    b) Find the area contained between the graphs of $\displaystyle y = \sqrt{3}cosx$ and $\displaystyle y = sinx$ between the points A and B.

    My problem is not in finding the solution, but in how the solution is obtained. For some reason, to find the area, you simply find the integral of $\displaystyle sinx - {sqrt}3cosx$ between the points B ($\displaystyle x = \frac{4\pi}{3}$) and A ($\displaystyle x = \frac{\pi}{3}$). I would have thought that you would need to find two seperate integrals, one to find the area enclosed between A and $\displaystyle \frac{\pi}{2}$ (i.e. the integral of $\displaystyle sinx - \sqrt{3}cosx$ between $\displaystyle x = \frac{\pi}{2}$ and $\displaystyle x = \frac{\pi}{3}$) and another to find the area betwee B and $\displaystyle \frac{\pi}{2}$ (i.e. the integral of $\displaystyle \sqrt{3}cosx - sinx$ between $\displaystyle x = \frac{4\pi}{3}$ and $\displaystyle x = \frac{\pi}{2}$). I would have thought that since part of the graph is below the x-axis then the area would be partly negative and therefore would counterract some of the area enclosed above the x-axis.

    Why is it that you can find the area, which is $\displaystyle 4 units^2$, using only a single integral, instead of two?
    Last edited by Flay; Sep 23rd 2008 at 12:07 AM.
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  2. #2
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    Quote Originally Posted by Flay View Post
    I know the solution to this particular question, but I'm not too sure of the reasoning behind it. The question asks;

    Two curves, $\displaystyle y = \sqrt{3}cosx$ and $\displaystyle y = sinx$, are drawn on a graph. The first two intersections to the right of the y-axis are labelled A and B.

    a) Solve the equation $\displaystyle \sqrt{3}cosx$ = $\displaystyle sinx$ to find the x-coordinates of A and B.

    b) Find the area contained between the graphs of $\displaystyle y = \sqrt{3}cosx$ and $\displaystyle y = sinx$ between the points A and B.

    My problem is not in finding the solution, but in how the solution is obtained. For some reason, to find the area, you simply find the integral of $\displaystyle sinx - {sqrt}3cosx$ between the points B ($\displaystyle x = \frac{4\pi}{3}$) and A ($\displaystyle x = \frac{\pi}{3}$). I would have thought that you would need to find two seperate integrals, one to find the area enclosed between A and $\displaystyle \frac{\pi}{2}$ (i.e. the integral of $\displaystyle sinx - \sqrt{3}cosx$ between $\displaystyle x = \frac{\pi}{2}$ and $\displaystyle x = \frac{\pi}{3}$) and another to find the area betwee B and $\displaystyle \frac{\pi}{2}$ (i.e. the integral of $\displaystyle \sqrt{3}cosx - sinx$ between $\displaystyle x = \frac{4\pi}{3}$ and $\displaystyle x = \frac{\pi}{2}$). I would have thought that since part of the graph is below the x-axis then the area would be partly negative and therefore would counterract some of the area enclosed above the x-axis.

    Why is it that you can find the area, which is $\displaystyle 4 units^2$, using only a single integral, instead of two?
    If f(x) > g(x) over [a, b] then the area bounded by the curves y = f(x) and y = g(x) between x = a and x = b is equal to $\displaystyle \int_a^b f(x) - g(x) \, dx$.

    Since $\displaystyle \sin x > \sqrt{3} \cos x$ over $\displaystyle \left[ \frac{\pi}{3}, \, \frac{4 \pi}{3} \right]$ the bounded area is $\displaystyle \int_{\pi/3}^{4 \pi/3} \sin x - \sqrt{3} \cos x \, dx$.
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