The Definite Integral The definite integral generalizes the concept - PowerPoint PPT Presentation

The Definite Integral The definite integral generalizes the concept of area.

The Definite Integral The definite integral generalizes the concept of area. Let’s begin with the physical intuition of this idea.

The Definite Integral The definite integral generalizes the concept of area. Let’s begin with the physical intuition of this idea. Let’s say the distance traveled by particle is given by the function:

The Definite Integral The definite integral generalizes the concept of area. Let’s begin with the physical intuition of this idea. Let’s say the distance traveled by particle is given by the function: s ( t ) = t 2 2

The Definite Integral The definite integral generalizes the concept of area. Let’s begin with the physical intuition of this idea. Let’s say the distance traveled by particle is given by the function: s ( t ) = t 2 2 s t

The Definite Integral If we take the derivative of this function:

The Definite Integral If we take the derivative of this function: s ′ ( t ) = t

The Definite Integral If we take the derivative of this function: s ′ ( t ) = t We get another function of time:

The Definite Integral If we take the derivative of this function: s ′ ( t ) = t We get another function of time: V t

The Definite Integral If we take the derivative of this function: s ′ ( t ) = t We get another function of time: V t This is velocity!

The Definite Integral Let’s say that, given velocity, we want the distance traveled over a period of time.

The Definite Integral Let’s say that, given velocity, we want the distance traveled over a period of time. Let’s begin with a very small time interval dt :

The Definite Integral Let’s say that, given velocity, we want the distance traveled over a period of time. Let’s begin with a very small time interval dt : V t

The Definite Integral Let’s say that, given velocity, we want the distance traveled over a period of time. Let’s begin with a very small time interval dt : V t

The Definite Integral Let’s say that, given velocity, we want the distance traveled over a period of time. Let’s begin with a very small time interval dt : V t dt

The Definite Integral Let’s say that, given velocity, we want the distance traveled over a period of time. Let’s begin with a very small time interval dt : V t dt

The Definite Integral Let’s say that, given velocity, we want the distance traveled over a period of time. Let’s begin with a very small time interval dt : V ≈ constant! t dt

The Definite Integral Now, if we consider that velocity is constant in this little interval, it is easy to calculate the distance traveled:

The Definite Integral Now, if we consider that velocity is constant in this little interval, it is easy to calculate the distance traveled: ds = Vdt

The Definite Integral Now, if we consider that velocity is constant in this little interval, it is easy to calculate the distance traveled: ds = Vdt V ≈ constant! t dt

The Definite Integral Now, if we consider that velocity is constant in this little interval, it is easy to calculate the distance traveled: ds = Vdt V ≈ constant! t dt

The Definite Integral Now, if we consider that velocity is constant in this little interval, it is easy to calculate the distance traveled: ds = Vdt V ≈ constant! Vdt t dt

Longer Time Intervals What if we want to find the distance traveled during a longer time interval?

Longer Time Intervals What if we want to find the distance traveled during a longer time interval? V t

Longer Time Intervals What if we want to find the distance traveled during a longer time interval? V t

Longer Time Intervals What if we want to find the distance traveled during a longer time interval? V t t 0

Longer Time Intervals What if we want to find the distance traveled during a longer time interval? V t t 0

Longer Time Intervals What if we want to find the distance traveled during a longer time interval? V t t 0

Longer Time Intervals What if we want to find the distance traveled during a longer time interval? V t t 0

Longer Time Intervals What if we want to find the distance traveled during a longer time interval? V t t 0

Longer Time Intervals What if we want to find the distance traveled during a longer time interval? V t dt t 0

Longer Time Intervals

Longer Time Intervals How can we improve this approximation?

Longer Time Intervals How can we improve this approximation? Idea:

Longer Time Intervals How can we improve this approximation? Idea: V t t 0

Longer Time Intervals How can we improve this approximation? Idea: V t t 0

Longer Time Intervals How can we improve this approximation? Idea: V t t 0

Longer Time Intervals How can we improve this approximation? Idea: V t t 0 As the number of rectangles approach ∞ , the sum of the areas approach the area below the curve!