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What if we wanted to find the area between the X axis and the function ... |
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F(x) = sinX + cosX |
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| and we want all of the area from X = 0 to X = p radians. | ||
| (for those not fluent in radian-ese, that's from 0 to 180 degrees) | ||
| The first thing we need to do is figure out if the function is always positive, | ||
| or always negative, or crosses the axis, or what. | ||
| When X = 0, cosX = 1 and sinX = 0. | ||
| So the total function value is 1. | ||
| That means our function starts out ABOVE the X axis | ||
| Now lets see if the function ever touches the X axis? | ||
| That is, is there a point where: | ||
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sinX + cosX = 0 |
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| Let's find out ... | ||
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sinX + cosX = 0 |
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sinX = - cosX |
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| In quadrant II (90 to 180), sine is positive and cosine is negative. | ||
| The sine and cosine are the same (except maybe for sign on the number) | ||
| at 45, 135, 215, 305 (45 + 90 x n). | ||
| The version of that in quadrant II is 135, also known as 3p/4 radians. | ||
| So we have two sections. One from 0 to 3p/4 radians | ||
| and one from 3p/4 radians to p radians. | ||
| Since the graph line is above the X axis at X = 0, | ||
| we know the first section is ok. That is, that area will turn out positive. | ||
| How do we tell if the second section will be positive or negative? | ||
| Take any point in that section (excluding 3p/4 ), | ||
| and test to see if the function has a positive or negative value. | ||
| So let's test the function at p. | ||
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F(x) = sinX + cosX |
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F(p) = sinp + cosp |
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F(p) = 0 - 1 |
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F(p) = - 1 |
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| So the section of the curve to the right of 3p/4 is below the X axis. | ||
| The graph looks something like this: | ||
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| So the problem works out this way ... | ||
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| That didn't hurt all THAT much did it? | ||
| Well at least it's over ... | ||
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copyright 2005 Bruce Kirkpatrick |
