Calculus Rhapsody

 Is this x defined?
    Is f continuous?
    How do you find out?
    You can use the limit process.
    Approach from both sides,
    The left and the right and meet.
    I'm just a limit,
    defined analytically
    Function's continuous,
    There's no holes,
    No sharp points,
    Or asymptotes.
    Anyway this graph goes
    It is differentiable for me ...
    for me.

    All year, in Calculus
    We've learned so many things
    About which we're going to sing.
    We can find derivatives
    And Integrals
    And the area enclosed between two curves.
    y' ... oooh
    Is the derivative of y
    Y equals x to the n,
    dy/dx equals n times x to the n - 1. 
    
    Other applications
    Of derivatives apply
    If y is divided or multiplied
    You use the quotient
    And product rules
    And don't you forget to do the dance ...
    Also ... oooh
    (don't forget the chain rule).
    Before you are done,   
    You gotta remember
    to multiply by the chain.

    I need to find the area under a curve
    Integrate! Integrate!
    You can use the integration.
    Raise the exponent by one
    Multiply the reciprocal.
    Add a constant. Add a constant.
    Add a constant. Add a constant.
    Add a constant labeled C.
    Labeled C.
    
    I'm just a constant, nobody loves me.
    He's just a constant
    might as well just call it C.
    Never forget to add the constant C.
    Can you find the area between f and g?
    Integrate f and then integrate g.
    Then subtract.

    To revolve around the y-axis
    Integrate outer radius
    minus inner radius squared.
    (multplied) multiplied by pi (multiply).
    Multiply the integral by pi!
    Pi tastes real good with whipped cream!
    Mama mia, mama mia!
    Mama mia, let me go!
    Precalculus did not help me prepare for
    Calculus, for Calculus, help me!
    
    So you think you can find out the limit of y?
    So you think you'll find zero and have it defined?
    Oh baby ... can't define that point baby.
    It's undefined ...
    Goes to positive and negative infinity.

    (Oooh yeah, Oooh yeah) 
    Differentiation, anyone can see
    Any mere equation
    It is differentiable for me.
    Any way this graph goes.