Verified answer

First, graph it:

https://www.desmos.com/calculator/uvrdos8wko

Now find the intersections:

y = x^2

y = (y^2)^2

y = y^4

0 = y^4 - y

0 = y (y^3 - 1)

y = 0 or 1

Points of intersection are (0, 0) and (1, 1).

Since the resulting solid is hollow (sort of donut-shaped), we can use washer method or shell method.

Using washer method, we slice the volume into a vertical stack of discs. Each disc has an outer radius of √y + 3 and an inner radius of y^2 + 3, as well as a thickness dy.

So:

dV = π [ (√y + 3)^2 - (y^2 + 3)^2 ] dy

dV = π (y + 6√y + 9 - y^4 - 6y^2 - 9) dy

dV = π (y + 6√y - y^4 - 6y^2) dy

Adding up all the discs from y=0 to y=1:

V = ∫ dV

V = ∫₀¹ π (y + 6√y - y^4 - 6y^2) dy

V = π (y/2 + 4y^(3/2) - 1/5 y^5 - 2y^3) |₀¹

V = π (1/2 + 4 - 1/5 - 2)

V = 2.3π

y=x^2 ---> x=sqrt(y) (outer radius)

x=y^2 ----> inner radius

y=(y^2)^2

y=y^4

y^4-y=0

y(y^3-1)=0

y=0

y=1

https://gyazo.com/fb9086c692f29f4b95b4b6e907a3602f

x=-3 is a vertical line so the rotation is along the y-axis

Volume = pi ∫ (sqrt(y)+3)^2 -(y^2+3)^2 ) ) dy

Integrate from 0 to 1

Volume = pi ∫ (-y^4-6y^2+y+6sqrt(y)) dy

= 2.3pi

By method of washers

We split the region to be rotated into a holysh!ttillion thin slices that are PERPENDICULAR to the axis of rotation.

Since x=-3 is vertical, the slices will have to be horizontal. And so each slice will be of with "dy".

The length of each slice is along x-axis, so we have to switch all the curves to "x=..."

x = y^2... okay

y = x^2... x = √y

Each of those slices when rotated swipes out a really thin circular disc with a hole in the middle around the line x=-3.

The outer edge of the disc has radius connected to the "upper" (farther from the line of rotation) function, √y + 3

The hole has radius connected to the "lower" (closer to line of rotation) function, y^2 + 3

So the area of each disc minus the hole is pi[ (√y + 3)^2 - (y^2 + 3)^2 ].

The volume of each of those rotated bits is then area * width = pi[ (√y + 3)^2 - (y^2 + 3)^2 ]dy.

Integrate from y=0 to y=1 because those are y-coordinates inside which the region is bounded.

By method of shells

The slices are PARALLEL to the axis of rotation, so we will have vertical slices like "y = ..."

y = x^2... okay

x = y^2... y = √x

Some slice has height (upper - lower) = √x - x^2, and width dx.

The slices when rotated make thin cylindrical shells of radius x+3 around the line of rotation.

Then the perimeter of each such shell is 2pi(x+3), and with height (√x - x^2) and width dx, a slice has volume

dV = perimeter * height * width = 2pi(x+3)(√x - x^2)dx

Again integrate, this time from x=0 to x=1.

In any case the volume is 2.3pi.

See the source link for a graph.

It is perhaps simplest to use the cylindrical shell method for the integration.

.. V = 2π*˜∫[0, 1] (3 +x)*(√x -x^2)*dx

.. = 2π(2x^(3/2) +(2/5)x^(5/2) -x^3 -(1/4)x^4)) . . . . evaluated at x=1

.. = 2π(2 +2/5 -1 -1/4)

.. V = 2.3π ≈ 7.22566 . . . . cubic units