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Calculus/area of a regions bound by graphs

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Expert: - 10/9/2009


Question
I just want to know if I got the correct answer for the following:

Find the are of the region bounded by the graphs of y=x^2-1 and y=x.

Answer:  Area= 1.87

Answer
They would cross where x² - 1 = x, or where x² - x - 1 = 0.

The solution to this equation would be x = (1 ± √(1+4))/2 = (1±√5)/2.

A point in the middle would be x=0.

Here, it can be seen that y=x is on top since x > x²-1 when x=0.

The endpoints are given above.

The integral would be to integrate x - x² + 1.
The integral of that is x²/2 - x³/3 + x.
This can be rewritten as (-x/3 + 1/2)x + 1)x.  The way the value of x could be stored in the calculater and recalled each time as the formula is applied.

Now you might have a spreadsheet.  In this case, put =(1-sqrt(5))/2 and =(1+sqrt(5))/2 in A1 and A2.  In B1, put =(-A1/3+0.5)*A1+1)*A1.  Copy that to B2.

Not rounding at all, I get 1.863389981, which rounds to 1.86.
The two values of x were -0.618033989 and 1.618033989.
The two function values were -0.348361657 and 1.515028324.
The result was 1.863389981.

I don't round at all, but let the computer keep track of my numbers.
On a calculator, I always used the memory.

Or course, the area only has an error of some number around 1/100 and basically you're right.

Calculus

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