Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. x = 1−y^4, x = 0; about x = 1.
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To find the volume of the solid obtained by rotating the region bounded by the curves about the line x = 1, you can use the method of cylindrical shells.
First, let's graph the region bounded by the curves x = 1 - y^4 and x = 0:
Since x = 1 - y^4 is a vertical line, we can see that it intersects the x-axis at x = 1 and the y-axis at y = 0.
To rotate this region about the line x = 1, we will integrate with respect to y.
The radius of each cylindrical shell is the distance from the line x = 1 to the curve x = 1 - y^4. This distance is given by (1 - y^4) - 1 = -y^4.
The height of each cylindrical shell is the difference in y-values, which is dy.
To set up the integral, we need the limits of integration. The region is bounded by x = 1 - y^4 and x = 0. To find the limits in terms of y, we set each equation equal to x and solve for y:
1 - y^4 = 0
y^4 = 1
y = ±1
So, we will integrate from y = -1 to y = 1.
The volume of each cylindrical shell is given by the formula: V = 2πrh, where r is the radius and h is the height.
Therefore, the volume of the solid can be found by integrating the expression 2π(-y^4)dy from y = -1 to y = 1:
V = ∫[from -1 to 1] 2π(-y^4)dy
Evaluating this integral will give you the volume of the solid of revolution.
Note: Make sure to perform the calculations to get the final answer.