This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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To calculate the volume using the Shell Method about the x-axis, we need to integrate with respect to y. The formula for the shell method rotating about the x-axis is V = _c^d 2 |y| (x_R - x_L) dy. Step 1: Find the points of intersection of the two curves. The given equations are xy = -5 and y = x - 6. From the second equation, express x in terms of y: x = y + 6. Substitute this into the first equation: (y+6)y = -5 y^2 + 6y = -5 y^2 + 6y + 5 = 0 Factor the quadratic equation: (y+1)(y+5) = 0 This gives y = -1 or y = -5. Now find the corresponding x values: If y = -1, then x = -1 + 6 = 5. Intersection point: (5, -1). If y = -5, then x = -5 + 6 = 1. Intersection point: (1, -5). The limits of integration for y are from c = -5 to d = -1. Step 2: Express x in terms of y for both curves and identify x_R and x_L. From xy = -5, we get x = -(5)/(y). From y = x - 6, we get x = y + 6. To determine which function is x_R (right) and x_L (left), pick a test value for y between -5 and -1, for example y = -3: For x = -(5)/(y): x = -(5)/(-3) = (5)/(3) ≈ 1.67. For x = y + 6: x = -3 + 6 = 3. Since 3 > (5)/(3), the curve x = y+6 is to the right (x_R) and x = -(5)/(y) is to the left (x_L). So, x_R = y+6 and x_L = -(5)/(y). Step 3: Set up the integral for the volume using the Shell Method about the x-axis. Since the region is below the x-axis (where y is negative), the radius of the cylindrical shell is |y| = -y. V = _-5^-1 2 (-y) (x_R - x_L) dy V = _-5^-1 2 (-y) ( (y+6) - (-(5)/(y)) ) dy V = 2 _-5^-1 (-y) ( y+6 + (5)/(y) ) dy V = 2 _-5^-1 (-y^2 - 6y - 5) dy Step 4: Evaluate the integral. V = 2 [ -(y^3)/(3) - (6y^2)/(2) - 5y ]_-5^-1 V = 2 [ -(y^3)/(3) - 3y^2 - 5y ]_-5^-1 Evaluate at the upper limit y = -1: (-((-1)^3)/(3) - 3(-1)^2 - 5(-1)) = (-(-(1)/(3)) - 3(1) + 5) = (1)/(3) - 3 + 5 = (1)/(3) + 2 = (7)/(3) Evaluate at the lower limit y = -5: (-((-5)^3)/(3) - 3(-5)^2 - 5(-5)) = (-(-(125)/(3)) - 3(25) + 25) = (125)/(3) - 75 + 25 = (125)/(3) - 50 = (125 - 150)/(3) = -(25)/(3) Subtract the lower limit value from the upper limit value: V = 2 ( (7)/(3) - (-(25)/(3)) ) V = 2 ( (7)/(3) + (25)/(3) ) V = 2 ( (32)/(3) ) V = (64)/(3) The volume generated is (64)/(3). What's next?