Question

find the volume v of the solid obtained by rotating the region bounded by the given curves about the specified line. x = y2, x = 1 − y2; about x = 4

Answer 1

The volume of the solid obtained by rotating the given region about x=4 is found using the method of cylindrical shells and setting up an integral with the appropriate bounds and formula.

Step-by-step explanation:

The volume V of the solid formed by rotating the region bounded by the curves x = y2 and x = 1 - y2 around the line x = 4 can be found using the method of cylindrical shells. To set up the integral, you first draw the region and identify the bounds of integration, which are the points where the two curves intersect, i.e., where y2 = 1 - y2, which simplifies to y = ±1/√2. The integral is then set up using the formula for the volume of a shell, V = 2π ∫ (radius)(height) dy, where the radius is 4 - x because the rotation is about x = 4, and the height is the difference between the curves (1 - y2) - y2. The bounds of integration are from -1/√2 to 1/√2, resulting in the final integral V = 2π ∫-1/√21/√2 (4 - y2 - (1 - y2))(2y) dy.

Answer 2

The volume V of the solid formed by rotating the region bounded by x = y² and x = 1 − y² about x = 4 is calculated using the washer or disk method and integrating the volume of each washer over the interval of intersection of the two curves.

Step-by-step explanation:

The volume V of the solid obtained by rotating the specified region around the line x = 4 can be found using the washer or disk method. The given curves are x = y² and x = 1 − y². The rotation axis (x = 4) is vertical, hence we consider the volume of the solid as a series of washers with the inner radius being distance from x = y² to x = 4, and the outer radius being the distance from x = 1 − y² to x = 4.

To calculate this volume, we use integrals, where each washer's volume is π(outer radius² - inner radius²) Δy, and we integrate with respect to y within the bounds where the curves intersect. The algebraic sum for the radii squared in this integral takes into account the distance from each curve to the axis of rotation at x = 4. Ultimately, evaluation of this definite integral from the lower to upper bounds of y gives us the total volume V of the rotated solid.