Find the volume of the solid formed by rotating the region enclosed by
x=0, \ x=1, \ y=0, \ y= 8 +x^{2}
about the x-axis.
The solid will be rotated about the x-axis, so we can apply the disk method by which vertical slices of thickness dx are integrated.
Radius of each slice
= y(x)
Volume of each slice
= πr²dx
= πy(x)²dx
Total volume can be obtained by integrating from x=0 to 1
V=∫ πy(x)²dx
=∫ π (8+x²)² dx
=∫ π (64+16x+x²)dx
=π [64x + 8x² + x³/3] x=0 to 1
=π[64+8+1/3]
=217π/3
Check my work.
To find the volume of the solid formed by rotating the region about the x-axis, we can use the method of cylindrical shells.
First, let's start by graphing the region and visualizing it. The region is bounded by the x-axis and the curve y = 8 + x^2. It is a parabolic shape that opens upwards with the vertex at the origin.
Now, we need to find the limits of integration. The region is bounded by x = 0 and x = 1, so our limits of integration will be 0 and 1.
Next, let's consider a thin vertical strip of width Δx at a distance x from the y-axis. The strip has a height equal to the difference in y-coordinates at the top and bottom of the region. The y-coordinate at the top is given by y = 8 + x^2, and the y-coordinate at the bottom is y = 0. So, the height of the strip is y = 8 + x^2 - 0 = 8 + x^2.
Now, we need to find the circumference of the cylindrical shell. The circumference of a cylinder is equal to 2πr, where r is the distance from the axis of rotation to the strip. In this case, r is equal to x, since the strip is a vertical distance x from the y-axis.
So, the circumference of the cylindrical shell is 2πx.
The thickness of the shell is Δx, so the volume of the cylindrical shell is given by the product of the circumference, height, and thickness:
Volumne of shell = 2πx * (8 + x^2) * Δx
To find the total volume, we need to sum up the volumes of all the cylindrical shells as x ranges from 0 to 1. This can be expressed as an integral:
Total volume = ∫(0 to 1) 2πx * (8 + x^2) dx
Evaluating this integral will give us the volume of the solid formed by rotating the region about the x-axis.