Give an equation representing the volume of the slice you would use in a Riemann sum representing the volume of the region. Then write a definite integral representing the volume of the region and evaluate it exactly. (The region is a cone.)

Please help, I've attempted this question 5 times already, and I keep getting it wrong on the first equation.

Question

Give an equation representing the volume of the slice you would use in a Riemann sum representing the volume of the region. Then write a definite integral representing the volume of the region and evaluate it exactly. (The region is a cone.)

Please help, I've attempted this question 5 times already, and I keep getting it wrong on the first equation.

anonymous · Answer

Welcome To OpenStudy!!!!!  =D

anonymous · Answer

sir the link you have provided has just brung me to the home page what i suggest you do is take a screenshot of the equation and attach it as a file

anonymous · Answer

The problem is attached below.

yanasidlinskiy · Answer

Same here,
$\huge\cal\color{cyan}{Welcome~to~OpenStudy!!!!}$
The half-circle has equation x²+y²=49.
At height y, this means that x = sqrt(49-y²)
So the approximate volume of this slice is 2*√(49-y²)*10*Δy
Now try to make the Riemann-sum and convert to an integral...

anonymous · Answer

Where are you getting 49 from? The diameter of the largest circular face is given in the problem as 40, which would make it $$x^2+y^2=40$$

anonymous · Answer

And it's not a half circular slice, it's a full circle--as is shown in the attached picture, the region is a cone.

mathmale · Answer

Let's start over.  Take a good look at that solid cone.  Imagine taking a nice, sharp knife and cutting a circular slice (shaped like a tin can lid) from the solid.  The quantity in question here is the AREA of this slipe; it is Pi*r^2.

We need to take the given dimensions of the cone to determine a function in x that specifies the associated y-value, which is, in turn, the radius of the thin circular slice.

Squaring that function provides us with the "radius squared."  Multiplying that by Pi results in a formula for the volume of that individual slice at any x value between 0 and 60 cm.

Does this provide enough info for anyone to run with it and correctly set up a Riemann Sum for the volume of said cone?

yanasidlinskiy · Answer

Ohh. Yea. I messed up there..

mathmale · Answer

Just supposing that we have that necessary function that represents the radius.  Call it f.  Then the integral (not the Riemann Sum) for calculating the volume of this cone would be:$$V=\pi \int\limits_{0~cm}^{60~cm}[f(x)]^2*dx$$Yes, yes, I understand we weren't asked to set up a definite integral, but for anyone familiar with how Reimann Sums lead to definite intergrals, this info should be helpful in setting up said Riemann Sum.

dumbcow · Answer

Think of the cone as a line that revolves around the x-axis

|dw:1403758354107:dw|

mathmale · Answer

Thanks for contributing this drawing.  I'd suggest that you and Corialos discuss how this formula f(x)=y=r=(x/3) was obtained, because knowing this is essential to the solution of this problem.

dumbcow · Answer

the function for radius ... f(x) = x/3 
comes from equation of line from origin to point (60,20)
For every "x" value this will give the "y" value of line which corresponds to length of radius in cone

anonymous · Answer

Thanks very much, you all, I think I have enough to be getting on with.