Sketch the region bounded by the curves y = x^2, y = x^4.

1) Find the area of the region enclosed by the two curves;
2) Find the volume of the solid obtained by rotating the above region about the x-axis;
3) Find the volume of the solid obtained by rotating the above region about the horizontal line with
equation y = 1 .

The region bounded by the two curves is between x = -1 and x = +1. Plot the two curves and you will see why.

1) Integrate (x^2 - x^4)dx from x = -1 to x = 1

2) Integrate pi*y1^2 - pi*y2^2 dx
= pi*(x^4 - x^8)dx from x = -1 to x = 1.
y1(x) = x^2
y2(x) = x^4

3) Integrate pi[(1 - y2)^2 - (1 - y1)^2]
dx from -1 to +1

To find the area of the region bounded by the curves y = x^2 and y = x^4, you need to determine the points where these two curves intersect. Setting the two equations equal to each other, x^2 = x^4, you can solve for x:

x^4 - x^2 = 0.

Factoring out an x^2, you get x^2(x^2 - 1) = 0. This equation gives you two solutions: x = 0 and x = 1.

The region bounded between the curves is from x = 0 to x = 1. To find the area between the curves, you integrate the difference of the equations from x = 0 to x = 1:

Area = ∫[0 to 1] (x^2 - x^4) dx.

Evaluating this integral will give you the area of the region.

To find the volume of the solid obtained by rotating this region about the x-axis, you can use the method of disks or washers. The cross-section of the solid at any x-value will be a disk or a washer, depending on whether the region is enclosed or not.

For disks (when the region is enclosed), the volume of each disk is given by π * (radius)^2 * (height). Since the region is bounded by the curves y = x^2 and y = x^4, the radius of each disk is x^2, and the height is the difference between the two curves: x^2 - x^4.

The volume of each disk is given by dV = π * (x^2)^2 * (x^2 - x^4) dx.

Integrating this expression from x = 0 to x = 1 will give you the volume of the solid obtained by rotating the region about the x-axis.

For washers (when the region is not fully enclosed), you need to subtract the volume of a smaller disk from the volume of a larger disk. This means you need to find the outer radius and the inner radius.

To find the outer radius, consider the curve y = x^2 as the outer curve and the curve y = x^4 as the inner curve. The outer radius is x^2, and the inner radius is x^4. The height of each washer is still (x^2 - x^4).

The volume of each washer is given by dV = π * [(outer radius)^2 - (inner radius)^2] * (height) dx.

Integrating this expression from x = 0 to x = 1 will give you the volume of the solid obtained by rotating the region about the x-axis.

For finding the volume of the solid obtained by rotating the region about the horizontal line with equation y = 1, you need to use the method of cylindrical shells. Consider a vertical strip in the region. The width of the strip is dx, the radius of the shell is x, and the height is the difference between the line y = 1 and the curve y = x^2 or y = x^4.

The height of each shell is given by (1 - x^2) or (1 - x^4), depending on the curve being considered.

The volume of each shell is given by dV = 2π * x * (height) dx.

Integrating this expression from x = 0 to x = 1 will give you the volume of the solid obtained by rotating the region about the horizontal line y = 1.

Once you have set up and evaluated the appropriate integrals, you can calculate the area or volume using the fundamental principles of calculus.