Assignment:
Q1. Compute the volume of the solid formed by revolving the region bounded by y =x2 , y = 4-x2 about (a) the x-axis; (b) y = 4.
Q2. Compute the volume of the solid formed by revolving the region bounded by y = x2 ,x =y2 and about (a) the y-axis; (b) x = 1.
Q3. Sketch the region, draw in a typical shell, identify the radius and height of the shell, and compute the volume for the region bounded by y = x and y = -x revolved about x=1 .
Q4. Sketch the region, draw in a typical shell, identify the radius and height of the shell, and compute the volume for the region bounded by y = x2 and y = 0 , -1≤x≤1 revolved about x = 2.
Q5.Sketch the region, draw in a typical shell, identify the radius and height of the shell, and compute the volume for the region bounded by x2 + y2 = 2y , revolved about y = 4.
Q6. Use cylindrical shells to compute the volume of the region bounded by x = y2 and x = 4, revolved about y = 2.
Q7. Compute the arc length exactly.
y = 2ln(4 - x2) , 0≤x≤1
Q8. Set up the integral for the surface area of the surface of revolution, and approximate the integral with a numerical method.
y =sinx , o≤x≤π , revolved about the x-axis
Q9. Set up the integral for the surface area of the surface of revolution, and approximate the integral with a numerical method.
y = √x , 1≤x≤2, revolved about the x-axis
Q10. Evaluate the integral ∫π02x cosx dx
Q11. Evaluate the integral ∫cot2 x csc2x dx
Q12. Determine whether or not the integral is improper. If it is improper, explain why.
a)∫∞0 x2/5 dx
b) ∫2-2 3/x dx
c) ∫∞2 3/x dx
Q13. Find the indicated limit
lim(x→∞)?(ex/x4)
Provide complete and step by step solution for the question and show calculations and use formulas.