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 total area bounded by the curves y = x^3 and y = 7x - 6, we need to find the points of intersection between these two curves and the x-axis.
Since the curves intersect at three points (-3, -27), (1, 1), and (2, 8), these are the x-values where the curves intersect the x-axis.
Let's denote these three x-values as x1, x2, and x3, respectively:
x1 = -3
x2 = 1
x3 = 2
To find the area between the curves, we need to calculate the areas of the individual regions separately. These regions can be found by integrating the difference between the two functions over their corresponding intervals.
Region 1: Bounded by y = x^3, y = 7x - 6, and x1 ≤ x ≤ x2
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The area of this region can be calculated using the definite integral:
Area1 = ∫[x1, x2] [7x - 6 - x^3] dx
= ∫[x1, x2] (7x - 6 - x^3) dx
Region 2: Bounded by y = x^3, y = 7x - 6, and x2 ≤ x ≤ x3
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Similarly, the area of this region can be calculated using the definite integral:
Area2 = ∫[x2, x3] [7x - 6 - x^3] dx
= ∫[x2, x3] (7x - 6 - x^3) dx
To find the total area bounded by the curves, we sum up the areas of these two regions:
Total Area = Area1 + Area2
By evaluating the integrals, we can calculate the total area.