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The region bounded by the curves y = -x² + 13x - 40 and y = 0 is rotated about the x-axis. Find the volume V of the resulting solid by any method.

User Sobia
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1 Answer

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Final answer:

The volume of the solid formed by rotating the curve y = -x² + 13x - 40 around the x-axis is found by solving the integral π∫ ((-x² + 13x - 40)²) dx over the x-values where the curve intersects the x-axis.

Step-by-step explanation:

To find the volume of the solid of revolution formed by rotating the region bounded by the curve y = -x² + 13x - 40 and the x-axis about the x-axis, we can use the method of discs/rings. This is a calculus problem which involves setting up an integral.

First, we need to find the points of intersection between the curve and the x-axis to determine the limits of integration. Setting y to zero gives -x² + 13x - 40 = 0. Solving this quadratic equation yields the limits of integration as the roots of the equation.

Once the limits are found, we can use the disc method. The volume V is given by the integral from a to b of π multiplied by the square of the function defining the radius, which in this case is -x² + 13x - 40. The integral will be of the form V = π ∫ab ((-x² + 13x - 40)²) dx.

After setting up the integral, we evaluate it over the interval from the lower to the upper limit which gives the volume of the solid.

User Jillan
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