Find the volume of the solid obtained by rotating the region under the graph of the function f(x)=x^3/2 about the x-axis over the interval [1,4]
Vol = π∫ y^2 dx from x = 1 to 4
= π ∫ x^3 dx from 1 to 4
= π[(1/4 x^4] from 1 to 4
= π( 4^4/4 - 1/4)
= (255/4)π
or, using shells of thickness dy, you can get the volume with a curved boundaty by
v = ∫[1,8] 2πrh dy
where r=y and h=4-x
v = ∫[1,8] 2πy(4-y^(2/3)) dy = (243/4)π
Now, add to that the small cylinder of radius 1 and height 3, which lies under the line y=1, and has volume 3π, and you get
(243/4 + 12/4)π = (255/4)π
To find the volume of the solid obtained by rotating the region under the graph of the function f(x) = x^(3/2) about the x-axis over the interval [1, 4], we can use the method of cylindrical shells.
1. First, let's visualize the region under the graph. The graph of f(x) = x^(3/2) over the interval [1, 4] is a curve that starts at (1, 1) and ends at (4, 8). When this region is rotated about the x-axis, it forms a solid shape.
2. Next, let's consider an infinitesimally small vertical strip with width "dx" along the x-axis. This strip lies at a distance "x" from the y-axis and has a height equal to the function value f(x) = x^(3/2) at that point.
3. Now, let's find the circumference of the cylindrical shell formed by this strip. The circumference of a cylinder is given by the formula 2πr, where "r" is the distance of the strip from the axis of rotation. In this case, the distance "r" is equal to x.
Circumference = 2πr = 2πx
4. The height of the cylindrical shell is given by the function value f(x) = x^(3/2).
5. Therefore, the volume of the cylindrical shell is given by the product of its height and circumference:
Volume of shell = height * circumference = f(x) * 2πx = 2πx * x^(3/2)
6. To find the total volume, we need to integrate the volume of each cylindrical shell. The integral should cover the range of x values from 1 to 4:
Total volume = ∫[1, 4] 2πx * x^(3/2) dx
7. Evaluating this integral will give us the desired volume of the solid.
Note: If you're familiar with definite integration, you can calculate the value of the integral using appropriate techniques (such as substitution or integration by parts) to find the exact volume. Otherwise, you can use a calculator or software that can perform symbolic integration to simplify and evaluate the integral for you.