Verify Cauchy' s mean value theorem          and find c, if f(x) = e^x , g(x) = e^-x ,          x ∈ [a,b]

Find the surface area formed by revolving four - cusped hypocycloid (asteroid) x^2/3 + y^2/3 = a^2/3 about the     x-axis 

Find the surface generated by revolving the portion of the curve y² = 4 + x cut off by the straight line x = 2, about the        x - axis

Using Rolle's theorem, prove that there is no real a for which the equation        x² - 3x + a = 0 two different roots in [-1,1].

Find c of Lagrange's mean value theorem for the following function          x³ + x in [1,2].

Solve (1 + y²)dx = (tan^-1y - x)dy

Identify the nature of singular points of the differential equation                          
x²(x - 2)y" + (x - 1)y' + 2xy = 0 

For any two real numbers a and b         (a < b) prove that a²+ ab+ b² = 3c²,for some c ∈ (a,b) using Lagrange's mean value theorem.

Solve equation                                                       
y= 3x + log_e p

Using Maclaurin's series expand tan x up to the term containing x⁵

Find the volume of sphere                    x² + y² + z² =  a² using triple integrals.