Find the exact area below the curve y=x^5(3−x) and above the x-axis.
the curve lies above the x-axis on the interval [0,3].
So, the area is
∫[0,3] x^5(3-x) dx = ∫[0,3] 3x^5 - x^6 dx = 1/2 x^6 - 6x^5[0,3] = 729/14
To find the exact area below the curve y = x^5(3 - x) and above the x-axis, we need to integrate the function over the appropriate interval.
First, let's find the points where the curve intersects the x-axis by setting y = 0:
0 = x^5(3 - x)
This equation has two solutions: x = 0 and x = 3.
To find the area below the curve and above the x-axis, we'll integrate the function from x = 0 to x = 3:
Area = ∫[0 to 3] (x^5(3 - x)) dx
To solve the integral, we'll expand the function and integrate each term separately:
Area = ∫[0 to 3] (3x^5 - x^6) dx
Now, we can integrate each term using the power rule of integration:
∫(3x^5) dx = (3/6)x^6 + C1 = (1/2)x^6 + C1
∫(-x^6) dx = (-1/7)x^7 + C2
Now, we can substitute the limits of integration:
Area = [(1/2)(3^6) + C1] - [(1/7)(3^7) + C2]
Simplifying further:
Area = (1/2)(729) - (1/7)(2187)
Area = 364.5 - 312.43
Area ≈ 52.07
So, the exact area below the curve y = x^5(3 - x) and above the x-axis is approximately 52.07 square units.
To find the exact area below the curve y = x^5(3−x) and above the x-axis, you can use integration.
First, let's find the x-intercepts of the curve by setting y = 0:
0 = x^5(3−x)
This equation has two solutions: x = 0 and x = 3.
Now, integrate the function y = x^5(3−x) between these x-intercepts to find the area.
∫[0 to 3] x^5(3−x) dx
To calculate this definite integral, we can first expand the equation:
∫[0 to 3] (3x^5 - x^6) dx
Next, we can use the power rule of integration to integrate the terms:
∫[0 to 3] (3x^5 - x^6) dx = [3/6 * x^6 - 1/7 * x^7] evaluated from 0 to 3
Evaluating the integral at the upper and lower limits:
[3/6 * 3^6 - 1/7 * 3^7] - [3/6 * 0^6 - 1/7 * 0^7]
Simplifying:
[3/6 * 729 - 1/7 * 2187] - [0 - 0]
Calculating:
[364.5 - 312.43] - [0 - 0]
= 52.07
Therefore, the exact area below the curve y = x^5(3−x) and above the x-axis is 52.07 square units.