CALCULUS: Which of the following represents the volume of the solid formed by revolving the region bounded by the graphs of y =, y = 1, and x = 3, about the line x = 3? A. π * [27,1]∫ (3-∛y)² dy
B. π * ∛
C. None of these

Disk method it is, since both given options integrate with respect to y.

First find where the boundaries of the region intersect.

• y = x ³ and y = 1 intersect at (1, 1)

• y = x ³ and x = 3 intersect at (3, 27)

• y = 1 and x = 3 intersect at (1, 3)

So the bounded region is the set of points

{ (x, y) | 1 ≤ x ≤ 3 and 1 ≤ y ≤ x ³ }

But since we're integrating with respect to y, rewrite this set so that y has constant limits:

{ (x, y) | ∛y ≤ x ≤ 3 and 1 ≤ y ≤ 27 }

Pick some point y in the interval [1, 27] and construct a disk centered at the given axis of revolution (x = 3). Such a disk will have radius 3 - ∛y because x = ∛y is the horizontal distance from the y-axis to the curve y = x ³ and x = 3 is itself 3 units away from the y-axis. Its height will be some small change in y, call it ∆y. Then the volume of this disk is

π (3 - ∛y )² ∆y

Now do the same thing for every y in [1, 27] - infinitely many of them! - and make ∆y very small, such that ∆y → dy. The volume of the solid is the sum total of the volumes of these infinitely many disks, given by the integral

and so the answer is A (assuming the limits of integration are listed from upper to lower).

27

step-by-step explanation:

the multiples of 15 are 15, 30,45, 60, 75, 90, 105, 120, 135, 150, 165, 180, 195,   210, 225, 240, 255, 270, 285, 300, 315, 330, 345, 360, 375,   390, 405, 420, 435, 450, 465, 480, 495, 510, 525, 540, 555, 570, 585, 600, 615, 630, 645, 660, 675, 690, 705, 720, and 735.

the factors of 15 are 1, 3, 5, and 15.

hope that , fam!

-une, your fren : )