calculus question

Question

Alex and Veronica were discussing the definite integral ∫₀³ (x² − 1)dx. Alex said it represented the total area bounded by f ( x ) = x² - 1 and the x-axis , between x = 0 and x = 3. Veronica said the total area was larger. Who was right and why? Also calculate the area.

 

Help!?

Answer 1

The graph of f(x) = x²-1, when graphed from x=0 to x=3 starts at (0,-1), crosses the x axis at (1,0) and then continues on to (3,9). 

 

If you just evaluate the definite integral, the portion from x=0 to x=1 is underneath the x-axis and will come out negative.

The definite integral from x=1 to x=3 will be above the x-axis and be positive. So if you just added these together, you would not get the area between the x-axis and the curve.

 

What you need to do is take the absolute value of the negative portion to properly represent the area and then add this to the second area.

 

∫₀¹ x²-1 dx is 1/3 x³ - x from 0 to 1 which is (1/3-1) - (0-0) = -2/3. This can be interpreted as an area of 2/3 that is underneath the axis....but the area is 2/3.

 

The secon part is above the axis so that there is no problem with the sign. (1/3 *27 - 3) - (1/3-1) = 6-(-2/3) = 6 2/3

 

So your total area is 2/3 + 6 2/3 = 7 and 1/3.

 

Hope this helped