Question

Find the volume of the solid formed by revolving the region bounded by the graphs of y = x2, x = 4, and y = 1 about the y-axis.

a: 39 times pi divided by 2
b:225 times pi divided by 2
c: 735 times pi divided by 2
d: None of these

Answer 1

@Luigi0210 @poopsiedoodle @pooja195

Answer 2

@ganeshie8

Answer 3

@bibby

Answer 4

can u plz help? :) @ganeshie8

Answer 5

@hartnn

Answer 6

Alright.. Here is how I see it..
Well if you look at the graphs of all those functions:
http://www.wolframalpha.com/input/?i=y%3Dx%5E2%2C+y%3D1%2C+x%3D4
You can see that they create some 'triangle' between them.
Of course this isn't a real triangle, but this shape is told us to be rotated around the y axis.
Now, if you imagine this rotation you'd figure that the x=4 vertical line will wrap the y axis and create some form of a cylinder.
This cylinder is actually hollow on the inside because \(y=x^2\) limits it from within, so if we could calculate the cylinder's volume and subtract the internal volume \(y=x^2\) takes within it then we'll get what we want.

Calculating the cylinder is pretty easy. We know the base is 4 because the rotated x=4 line will make circles around the y-axis with radius of 4.
So \(b = \pi(4)^2 = \pi 16\).
The height of the cylinder is the height of this 'triangle'. We know it is bounded on bottom by y=1. On top it is bounded by \(y = x^2\). The intersection of x=4 and \(y = x^2\) is at \(y=4^2=16\).
So the height of the cylinder is \(h = 16 - 1 = 15\), and the cylinder's volume:
$$ b \cdot h = \pi 16 \cdot 15 = \pi 240 $$

Now we can calculate the volume that within \(y = x^2 \) and the y-axis using integration.
We can tell that when \(y = x^2\) is rotated around the y-axis it will also create circles. But this time, the radius of the circles changes. The radius of each circle is the \(x\) value of the point creating it.
If we step along the y-axis from 0 and up, we can get the x value for every y we step and use this x value as a radius of the circle \(y = x^2\) makes around the y-axis for that 'y'.
We can use the area of that circle to form a small cylinder, the circle is the base and the step side along the y-axis is the height.
As the step size gets smaller and smaller our cylinders total volume will describe more and more accurately the volume we're seeking.

Let \(dy\) be our infinitesimal step along the y-axis and knowing \(x^2 = y\). We can say that:
$$
\int \pi x^2 dy = \pi\int y dy = \pi \frac{y^2}{2}
$$The top limit of our shape is y=16, the bottom limit is y =1, so we want to calculate the volume inbetween:
$$
\text{inner_volume} = \pi \frac{y^2}{2} \bigg|^{16}_1 = \pi \frac{16^2}{2} - \pi \frac{1^2}{2} = \pi \frac{256 - 1}{2} = \frac{\pi255}{2}
$$
Subtracting from the cylinder's volume:
$$
\text{cylinder_volume} - \text{inner_volume} = \pi 240 - \frac{\pi255}{2} = \\
= \frac{\pi480 - \pi255}{2} = \frac{\pi 225}{2}
$$
Hope it helps

Answer 7

thank u !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Answer 8

Sure np