Final answer:
To find the volume of the solid bounded by the curves y = x² - 7 and y = -6, we can integrate the area of the square cross-sections perpendicular to the x-axis. By determining the limits of integration and calculating the integral, we find that the volume of the solid is 53/15 cubic units.
Step-by-step explanation:
To find the volume of the solid, we need to integrate the area of the cross-sections perpendicular to the x-axis. Since the cross-sections are squares, the area of each cross-section would be side length squared. To determine the limits of integration, we need to find the x-values where the curves y = x² - 7 and y = -6 intersect. Setting these equations equal to each other gives us x² - 7 = -6, which simplifies to x² = 1. Solving for x, we find x = ±1. Therefore, the limits of integration are -1 and 1.
Now, we can integrate the area of the square cross-sections from -1 to 1. The side length of each square would be the difference between the two curves: (x² - 7) - (-6) = x² - x - 1. Thus, the volume of the solid can be calculated as follows:
V = ∫[from -1 to 1] (x² - x - 1)² dx
Next, we expand and simplify the expression (x² - x - 1)²:
V = ∫[from -1 to 1] (x⁴ - 2x³ + 2x² + 2x + 1) dx
Integrating each term, we get:
V = [1/5x⁵ - 1/2x⁴ + 2/3x³ + x² + x] [from -1 to 1]
Substituting the limits of integration:
V = [(1/5(1)⁵ - 1/2(1)⁴ + 2/3(1)³ + 1² + 1) - (1/5(-1)⁵ - 1/2(-1)⁴ + 2/3(-1)³ + (-1)² + (-1)]
Simplifying the expression inside the brackets:
V = [(1/5 - 1/2 + 2/3 + 1 + 1) - (-1/5 - 1/2 - 2/3 + 1 + 1)]
Calculating the values inside the brackets:
V = [(47/30) - (-59/30)]
Simplifying:
V = 106/30 = 53/15
Therefore, the volume of the solid is 53/15 cubic units.