To solve this problem, we will determine two key quantities:
1. The area under the curve [tex]\( y = \frac{1}{x^4 + 1} \)[/tex] from [tex]\( x = 0 \)[/tex] to [tex]\( x = 1 \)[/tex].
2. The volume of the solid formed by revolving this area around the [tex]\( y \)[/tex]-axis.
### Step 1: Finding the Area Under the Curve
The area [tex]\( A \)[/tex] under the curve [tex]\( y = \frac{1}{x^4 + 1} \)[/tex] from [tex]\( x = 0 \)[/tex] to [tex]\( x = 1 \)[/tex] is given by the definite integral of the function with respect to [tex]\( x \)[/tex]:
[tex]\[
A = \int_{0}^{1} \frac{1}{x^4 + 1} \, dx
\][/tex]
The result of evaluating this integral numerically is:
[tex]\[
A = 0.8669729873399111
\][/tex]
### Step 2: Finding the Volume of Revolution
To find the volume of the solid formed by revolving the given region around the [tex]\( y \)[/tex]-axis, we use the method of disks. The volume formula, when revolving around the [tex]\( y \)[/tex]-axis, is given by:
[tex]\[
V = 2\pi \int_{x_{\text{lower}}}^{x_{\text{upper}}} x \cdot f(x) \, dx
\][/tex]
However, based on the previous calculation, we can also express the volume formula in a more straightforward way. The volume of revolution around the [tex]\( y \)[/tex]-axis for this function is given by multiplying the area under the curve by [tex]\(\pi\)[/tex]:
[tex]\[
V = \pi \times A
\][/tex]
Substituting the computed area:
[tex]\[
V = \pi \times 0.8669729873399111 \approx 2.7236759678878615
\][/tex]
### Summary
- The area under the curve [tex]\( y = \frac{1}{x^4 + 1} \)[/tex] from [tex]\( x = 0 \)[/tex] to [tex]\( x = 1 \)[/tex] is approximately [tex]\( 0.8669729873399111 \)[/tex].
- The volume of the solid formed by revolving this area around the [tex]\( y \)[/tex]-axis is approximately [tex]\( 2.7236759678878615 \)[/tex].
Thus, the final answers are:
- Area under the curve: [tex]\( 0.8669729873399111 \)[/tex]
- Volume of the solid of revolution: [tex]\( 2.7236759678878615 \)[/tex]
