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.

(Upper limit b, lower limit a) ∫ba 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!

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

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

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.

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

MINOAS

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.