Assignment:
Q1. A paraboloid dish (cross section y = x2 ) is 8 units deep. It is filled with water up to a height of 4 units. How much water must be added to the dish to fill it completely?
Q2. Write an integral that represents the volume of the solid formed by rotating the region bounded by y = f(x) , x = a , x = b , and y = 0 about (a) the x axis; (b) the line x = b ; and (c) the line x = c, where c>a, b.
Q3. Consider the solid formed by rotating the curve y = f(x) from x = a to x= b about the x axis. Let V(x) be the function whose value is the volume of the solid between x= a and x =x .
(a) What is V(a)?
(b) For some small Δx , what is V( x + Δx)?
(c) Using the definition of the derivative, find the definite integral that represents the total volume of the solid. (Let F(x) be a function such that dF/dx = π[f(x)]2 .)
Q4. Find the volume of the top quarter of a sphere: ∫rr/2 πx2dy .
Provide complete and step by step solution for the question and show calculations and use formulas.