The curve y=x^3 intersects the line y=7x-6 at three points, (-3,-27), (1,1), and (2,8). Find the total area bounded by y=x^3 and y=7x-6.
To find the area bounded by two curves, you need to subtract the smaller curve from the larger curve and integrate the result over the interval where it is bounded.
First, let's find the intersection points between the two curves y = x^3 and y = 7x - 6 by setting them equal to each other:
x^3 = 7x - 6
Rearranging the equation, we get:
x^3 - 7x + 6 = 0
This can be factored as:
(x - 1)(x + 2)(x - 3) = 0
So, we have three intersection points: x = 1, x = -2, and x = 3.
To find the area, we need to integrate the difference between the two curves from the smallest root to the largest root. Since the cubic curve is above the line, we subtract the line from the curve.
The equation for the area is:
A = ∫[a,b] (f(x) - g(x)) dx
where a and b are the x-coordinates of the intersection points and f(x) is the cubic curve (x^3) and g(x) is the line (7x - 6).
Let's calculate the area between x = -2 and x = 1 first:
A1 = ∫[-2,1] (x^3 - (7x - 6)) dx
Simplifying, we get:
A1 = ∫[-2,1] (x^3 - 7x + 6) dx
To integrate the function x^3 - 7x + 6, we need to find the antiderivative:
A1 = [1/4x^4 - 7/2x^2 + 6x] from -2 to 1
Evaluating the antiderivative at the upper and lower limits and subtracting, we get:
A1 = [(1/4 * 1^4 - 7/2 * 1^2 + 6 * 1) - (1/4 * (-2)^4 - 7/2 * (-2)^2 + 6 * (-2))]
Simplifying, we get:
A1 = [1/4 - 7/2 + 6 + 1/4 - 7/2 - 12]
A1 = -13/4
Next, we calculate the area between x = 1 and x = 3:
A2 = ∫[1,3] (x^3 - (7x - 6)) dx
Similarly, we integrate the function x^3 - 7x + 6:
A2 = [1/4x^4 - 7/2x^2 + 6x] from 1 to 3
Evaluating the antiderivative at the upper and lower limits and subtracting, we get:
A2 = [(1/4 * 3^4 - 7/2 * 3^2 + 6 * 3) - (1/4 * 1^4 - 7/2 * 1^2 + 6 * 1)]
Simplifying, we get:
A2 = [81/4 - 63/2 + 18 - 1/4 + 7/2 - 6]
A2 = 47/4
Now, we can find the total area by summing A1 and A2:
Total Area = A1 + A2 = -13/4 + 47/4 = 34/4 = 8.5
Therefore, the total area bounded by y = x^3 and y = 7x - 6 is 8.5 square units.