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
f(x)=(1-x)*x^9
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.
http://img411.imageshack.us/img411/1397/1296510221.png
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.
To find the exact area below the curve (1-x)*x^9 and above the x-axis, we will need to integrate the function over the desired interval.
The given function is (1-x)*x^9.
First, we need to find the points where the curve intersects the x-axis. Since the function is (1-x)*x^9, the curve will intersect the x-axis at x = 0 and x = 1.
Next, we need to determine the interval over which we want to find the area. In this case, we want to find the area below the curve and above the x-axis, so the interval will be from x = 0 to x = 1.
Now we can set up the definite integral to find the exact area:
∫[0,1] (1-x)*x^9 dx
To solve this integral, we can use integration rules and techniques:
Step 1: Expand the function
(1-x)*x^9 = x^9 - x^10
Step 2: Integrate each term separately
∫[0,1] (x^9 - x^10) dx = (∫[0,1] x^9 dx) - (∫[0,1] x^10 dx)
Step 3: Integrate each term using the power rule:
∫[0,1] x^9 dx = (1/10)x^10 | [0,1] = (1/10)(1^10 - 0^10) = 1/10
∫[0,1] x^10 dx = (1/11)x^11 | [0,1] = (1/11)(1^11 - 0^11) = 1/11
Step 4: Calculate the difference between the integrals:
(1/10) - (1/11) = 1/110
Therefore, the exact area below the curve (1-x)*x^9 and above the x-axis is 1/110 square units.