Hey. I'm Chloe:) Please help with some integration problems???

1. A glass vase has the shape of the solid obtained by rotating about the y–axis the area inthe first quadrant lying over the x–interval [0,a] and under the graph of y = x^2.

Determine how much glass is contained in the vase.

2. Is the following statement true or false? Justify your answer.

(Upper limit b, lower limit a) ∫^{b}_{a} f(x) dx ≤ 0 implies f (x)≤ 0 on [a,b]

3. Solve the differential equation (dy/dx) = (4xy) / (1+x^2)

4. Find the unique function y(x) satisfying the differential equation with initial condition

(dy/dx) - (x^2)y = 0, y(1) = 1

Thank you. Any Help with the questions is greatly appreciated!

Re: Hey. I'm Chloe:) Please help with some integration problems???

Chloe

1. find the volume of the glass by calculating the integral π(ydy) from 0 to a^2.

2. No it is not correct .The integral f(x) from a to b being < 0 simply means that the total area + and - gives a negative result but the function f(x) might be positive

3. separate the variables and get dy/y =4x/(1+x^2) then integrate....it is easy

4. separate the variables again and find dy/y=x^2dx then integrate and use x=1 ,y=1 to find the constant C of integration.

good luck

Re: Hey. I'm Chloe:) Please help with some integration problems???

Would this working for qn 1 be correct?

V = ∫(x = 0 to a) 2πx * x^2 dx

..= ∫(x = 0 to a) 2πx^3 dx

..= 2πx^4/4 {for x = 0 to a}

..= (π/2)a^4.

It looks similar to what you've mentioned but its w.r.t. x instead of y.

Also, what kind of functions would prove the statement false?

A modulus function is all i can think of, and i'm not too sure thats right either.

Thanks a lot for helping

Re: Hey. I'm Chloe:) Please help with some integration problems???

Chloee

yes the answer is ..= (π/2)a^4.

but when you compute volumes by rotating curves over the y-axis you must get directly the integral πx^2dy from a to b ( the limits of integration ,in your case 0 and a^2.

download this file for a quick revision on this subject.

http://www.mathcentre.ac.uk/resource...mes-2009-1.pdf

MINOAS

Re: Hey. I'm Chloe:) Please help with some integration problems???

The first problem, "Determine how much glass is contained in the vase", does not, to me, seem to ask for the volume of the figure! A vase, is, after all, hollow so that you can put things, like flowers, in it. It seems to me it should be a surface area multiplied by the thickness of the glass- but we are not told any thickness.

Re: Hey. I'm Chloe:) Please help with some integration problems???

But what we're calculating is the volume of the figure isn't it? All we're doing is just subbing it into the formula for volume...

For the second question, is there an example of functions that prove the statement false?

Re: Hey. I'm Chloe:) Please help with some integration problems???

I'd like to point out to the OP that you should not post any more than two questions per thread, and you must show that you have made an effort for each question by posting your working and pointing out exactly where you are stuck.

I completely agree with HallsofIvy that the first question is ambiguous. However, the amount of glass used WOULD be a volume. The trouble is that we don't know the THICKNESS of the glass, which means we can not tell how much volume needs to be subtracted from the total solid volume you have found in order to hollow it out. I suggest you speak to your lecturer to determine exactly what is expected in this question.