FTC and Antiderivatives

Math 111

That last term from the applet...

Why did it change?

+C

Why the +C from the general antiderivative?

We know FTC

Part 1: If \( v(t) \) is a velocity function, we can write \( p(t) \) as: \[ p(t)=\int_0^t v(x)dx \] Why the variables?

Part 2: If \( F'(x)=f(x) \), \[ \int_a^b f(x)dx = F(b)-F(a) \]

Notation

We use \( \int f(x) dx \) to denote the general antiderivative to \( f(x) \)

Scenarios

  • Coal Scrubbers: Pollution rate * \( \Delta \) Time
  • Mars Rover: Dust rate * \( \Delta \) Distance
  • Hydrostatic: Pressure * Width * \( \Delta \) Height
  • Chicken: Distance rate * \( \Delta \) Time

Scenarios Units

  • Coal Scrubbers: Tons/Month * \( \Delta \) Months
  • Mars Rover: Mg/km * \( \Delta \) km
  • Hydrostatic: Lbs/ft 2 * ft * \( \Delta \) ft
  • Chicken: miles/hour * \( \Delta \) Hours

Using the FTC

Evaluate:

\[ \int_{-1}^2 \sin(x) dx \]

\[ \int_{-1}^2 \cos(x) dx \]

\[ \int_{-1}^t \sin(x) dx \]

\[ \int_3^4 \dfrac{1}{x} dx \]

General propoerties

\[ \int_a^b f(x) dx = -\int_b^a f(x) dx \]

\[ \int_a^b f(x) dx + \int_b^c f(x) dx = \int_a^c f(x) dx \]

What if \( a < c < b \)?

Challenge

Find a general antiderivative to

\[ f(x)=e^{5x^2} \]

Notation

\[ \int \dfrac{1}{x} dx \] vs \[ \int \dfrac{1}{t} dt \] vs \[ \int \dfrac{1}{(cabin)} d(cabin) \]

Integral Rules

Note that the FTC means these are useful for both indefinite definite integrals.

  • Monomials
  • Polynomials
  • Trig function
  • log functions
  • exponential functions

Note: products and compositions are going to be harder

General strategies

  • If you have a polynomial divided by \( x^a \)
  • If you know a particular value, e.g. F(1)=4
  • Derivative of integral vs integral of derivative

Back to the applet

  • How do we know the constant term in \( q(t) \)?
  • How does it change when we change \( a \)? How can we figure out the value of the constant?