posted by Amanda
Find the exact area below the curve (1-x)*x^9 and above the x axis
Were you given (a,b) values?
From x = ? to ?
First you define the function as
Inspection of its 10 factors should indicate that the function crosses the x-axis at two points (0,0), (0,1).
The leading coefficient is -x^10, which means that the major part of the function is concave downwards, and is above the x-axis only between x=0 and 1.
Check: f(0.5)=(1-0.5)*0.5^9 > 0
The area sought is thus between the limits of x=0 and x=1.
The area below a curve f(x) is
∫f(x)dx between the limits of integration (0 to 1).
The function can be split up into a polynomial with two terms, and is thus easy to integrate.
Inspection of the graph between 0 and 1 and an approximate calculation of the area shows that the area should be in the order of 0.01. Post your answer for a check if you wish.
Here's a graph of the function between 0 and 1.
I just looked at the graph.
You want the area from x = 0 to x = 1.
| = integrate symbol
| x^9(1 - x)
| x^9 - x^10
Then plug in x, from 0 to 1.