Find the volume of the solid formed by revolving the region bounded by the graphs of y = x^2, x = 4, and y = 1 about the y-axis.
I'm supposed to use this equation right? piR^2-x^r^2. and then change y=x^2 to x=sqrty. That's all I got.
as you say, you have to think of the solid as a stack of washers, of thickness dy. So,
v = ∫[1,16] π(R^2-r^2) dy
where R = 4 and r = x = √y. So,
v = ∫[1,16] π(16-y) dy
= 225π/2
As a check on your answer, think of the solid as a set of nested cylinders of thickness dx. Then you have
v = ∫[1,4] 2πrh dx
where r = x and h = y-1 = x^2-1. Thus,
v = ∫[1,4] 2πx(x^2-1) dx
= 225π/2
225 pi/2 is right. i just did this question.
Well, let's break it down step by step, and I promise to make it as entertaining as possible!
First, we need to find the limits of integration. The region is bounded by x = 4 and y = 1.
Now, let's think about how we're going to revolve this region. We want to revolve it around the y-axis, which means all our formulas need to be in terms of y.
You're absolutely right! To express y = x^2 as x = sqrt(y), that's a smart move. So now, we have x = sqrt(y).
Now, let's consider the shape that we're revolving. We're dealing with a solid formed by revolving a region bounded by the parabola y = x^2, the line x = 4, and the line y = 1.
Imagine this solid as a fancy hat, with the y-axis as the center point of the hat. It's a hat for math enthusiasts!
Now, let's slice our hat into really thin cylindrical disks, or should I say hats. Each hat has a thickness dy.
To find the volume of each small hat, we use the formula for the volume of a cylinder: pi*r^2*h.
Our radius is the distance from the y-axis to the curve x = sqrt(y). Suppose the height of each hat is dy, then our radius is sqrt(y).
Now, we have everything we need for each small hat's volume: pi*(sqrt(y))^2*dy.
To find the total volume, we integrate this expression from y = 0 to y = 1, because the region is bounded by y = 1.
So, the integral becomes:
V = ∫[0,1] of (pi*(sqrt(y))^2) dy.
Now, let's do some math-fu! Simplify the expression inside the integral:
V = pi * ∫[0,1] y dy.
Integrating y with respect to y is pretty straightforward.
V = pi * [y^2/2] from 0 to 1.
Plug in the values:
V = pi * (1^2/2 - 0^2/2).
V = pi * (1/2).
And finally:
V = pi/2.
So the volume of this stylish mathematical hat is pi/2. I bet it'll look great on you!
To find the volume of the solid formed by revolving the region bounded by the graphs of y = x^2, x = 4, and y = 1 about the y-axis, we can use the method of cylindrical shells.
First, let's sketch the region bounded by the graphs:
y = x^2
/|
/ |
x = 4 / |
__/ |
/| /
/ | /
/ | /
/ | /
Notice that the region lies between the y-axis (x = 0), the curve y = x^2, the line x = 4, and the line y = 1.
Since we are revolving this region around the y-axis, the radius of each cylindrical shell will be the distance from the y-axis to the corresponding point on the curve y = x^2. This distance is simply x.
The height of each cylindrical shell will be the difference between the two functions: (4 - x^2) - 1 = 3 - x^2.
To calculate the volume of each cylindrical shell, we'll use the formula:
V = 2πrh
where r is the radius of the shell, and h is the height of the shell.
Substituting the values for r and h, we have:
V = 2πx(3 - x^2)
Now, let's find the limits of integration. Since the region is bounded by x = 4 on the right, and the y-axis on the left, our limits of integration for x are 0 to 4.
To find the total volume, we'll integrate this expression with respect to x over the given limits:
V = ∫(from 0 to 4) 2πx(3 - x^2) dx
Evaluating this integral, we have:
V = 2π ∫(from 0 to 4) (3x - x^3) dx
V = 2π [3/2 x^2 - 1/4 x^4] (from 0 to 4)
V = 2π [(3/2)(4)^2 - 1/4(4)^4) - ((3/2)(0)^2 - 1/4(0)^4)]
V = 2π [(3/2)(16) - 1/4(256) - 0]
V = 2π [(24) - (64)] = 2π (-40)
Therefore, the volume of the solid formed by revolving the given region about the y-axis is -80π (cubic units).
Note: The negative sign indicates that the final result is an "oriented" volume, meaning it has a direction in space. However, when discussing the magnitude of the volume, we usually omit the negative sign.
To find the volume of the solid formed by revolving the region bounded by the graphs, you can use the method of cylindrical shells. Here's how you can solve it step by step:
1. First, sketch the region bounded by the graphs y = x^2, x = 4, and y = 1. This will be the area that is going to be rotated around the y-axis.
2. To use the cylindrical shell method, we need to express each shell as a function of y instead of x. Since y = x^2, we can rewrite it as x = √y, which represents the right side of the region we are rotating.
3. Now, let's calculate the radius of each cylindrical shell. The radius (R) of each shell is simply the distance from the axis of rotation (y-axis) to the function x = √y. In this case, the radius (R) is equal to √y.
4. Next, we need to calculate the height (h) of each cylindrical shell. The height (h) is the difference between the y-values of the upper and lower boundaries of the region. In this case, it will be the difference between y = 1 (lower boundary) and y = x^2 (upper boundary). Therefore, it will be h = 1 - x^2.
5. Now, we have all the components required to calculate the volume of each cylindrical shell, which is given by V = 2πRh. Substitute the values of R and h into the equation, and we get V = 2π(√y)(1 - x^2).
6. Finally, we need to integrate the expression V = 2π(√y)(1 - x^2) with respect to y over the appropriate interval. Since y ranges from 0 to 1 in this case, the integral setup is as follows: ∫[0,1] 2π(√y)(1 - √y^2) dy.
After evaluating the integral, you will obtain the volume of the solid formed by revolving the region bounded by the graphs y = x^2, x = 4, and y = 1 about the y-axis.