Please help!!!
Given the equations Y=√x, y=3 and x=0
1.) find the volume that is formed by rotating the shaded region about the line y=4
2.) find the volume that is formed by rotating the shaded region about the line x=6
I have a test on this tomorrow and I could seriously use some help!
http://www.youtube.com/watch?v=IZ8W-h764Cc&feature=relmfu
Don't worry, I'm here to help you! Let's break down the problem step by step and solve it together.
First, let's visualize the shaded region so we have a better understanding of what we're working with. The equation y = √x represents a part of the curve of a square root function, and y = 3 and x = 0 are two horizontal and vertical lines, respectively.
To find the volume formed by rotating the shaded region about the line y = 4, we need to use the method of cylindrical shells. Here's how we can do it:
Step 1: Determine the limits of integration.
Since the curve y = √x and the horizontal line y = 3 intersect at y = 3, we need to find the corresponding x-coordinate. Setting √x = 3 and solving for x, we get x = 9. Hence, the limits of integration for x are 0 to 9.
Step 2: Determine the height function.
The height of each cylindrical shell will be the difference between y = 4 and y = √x. So the height function h(x) is given by h(x) = 4 - √x.
Step 3: Determine the radius function.
The radius of each cylindrical shell will be the x-coordinate itself. So the radius function r(x) is given by r(x) = x.
Step 4: Calculate the volume.
The volume V is given by the integral of the product of the height function, radius function, and the thickness of the shells. In this case, the thickness is dx (infinitesimally small change in x). Thus, the volume can be calculated as follows:
V = ∫(2πrh(x))dx
= ∫(2πx(4 - √x))dx
= 2π∫(4x - x√x)dx
= 2π[2x^2 - (2/3)x^(3/2)] evaluated from 0 to 9
By solving the definite integral and evaluating it between x = 0 to 9, you will find the volume formed by rotating the shaded region about the line y = 4.
Similarly, to find the volume formed by rotating the shaded region about the line x = 6, you need to follow the same steps. The only difference is that the limits of integration will now depend on the x-values where the curve y = √x intersects with the vertical line x = 6.
I recommend using a calculator or software that can perform symbolic integration to help you evaluate the definite integral.