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!