Find the exact area below the curve y=x5(2−x) and above the x-axis.
area =
The curve y=x^5(2−x) intersects the x-axis at x=0 and x=2. To find the area below the curve and above the x-axis, we can integrate the function from 0 to 2:
∫₀² x⁵(2-x) dx
Expanding the polynomial and simplifying, we get:
∫₀² (2x^5 - x^6) dx
Integrating term by term, we get:
[2/6*x^6 - 1/7*x^7] from 0 to 2
Plugging in the limits of integration, we get:
[2/6*(2^6) - 1/7*(2^7)] - [2/6*(0^6) - 1/7*(0^7)]
Simplifying:
1016/105
Therefore, the exact area below the curve y=x^5(2-x) and above the x-axis is 1016/105 square units.