Question

Find the exact area below the curve y=x^(3)(1 - x) and above the x-axis?

Answer 1

Integrating you get (x^4)/4-(x^5)/5 +c for the indefinite integral, and from a to b is just do the same substitution I did on the other exercise

Answer 2

y(x)=x^3(1 - x)
y(x)=x^3 - x^4

First, we have to find the two points at which the function intersects the x-axis. Let's evaluate the original equation to find where it is equal to zero.
y = 0
x^3(1-x)=0
x=0 and x=1
At (0,0) and (1,0)

Then Integrate the function, and evaluate the integral between those 2 points.
\[\int\limits_{0}^{1} y(x) dx = \int\limits_{0}^{1} (x^3-x^4) dx\]
= x^4/4-x^5/5
Area above x-axis = y(1)-y(0)
= (1/4-1/5)-(0-0)=1/20