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EGR 1101: Unit 11 Lecture # 1PowerPoint Presentation

EGR 1101: Unit 11 Lecture # 1

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EGR 1101: Unit 11 Lecture # 1

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EGR 1101: Unit 11 Lecture #1

Applications of Integrals in Dynamics: Position, Velocity, & Acceleration

(Section 9.5 of Rattan/Klingbeil text)

- Recall that differentiation and integration are inverse operations.
- Therefore, any relationship between two quantities that can be expressed in terms of derivatives can also be expressed in terms of integrals.

Position x(t)

Derivative

Integral

Velocity v(t)

Derivative

Integral

Acceleration a(t)

- Ball dropped from rest
- Ball thrown upward from ground level
- Position & velocity from acceleration (graphical)

- Recall that:
- Differentiating a parabola gives a slant line.
- Differentiating a slant line gives a horizontal line (constant).
- Differentiating a horizontal line (constant) gives zero.

- Therefore:
- Integrating zero gives a horizontal line (constant).
- Integrating a horizontal line (constant) gives a slant line.
- Integrating a slant line gives a parabola.

- The change in velocity between times t1 and t2 is equal to the area under the acceleration curve between t1 and t2:

- The change in position between times t1 and t2 is equal to the area under the velocity curve between t1 and t2:

EGR 1101: Unit 11 Lecture #2

Applications of Integrals in Electric Circuits

(Sections 9.6, 9.7 of Rattan/Klingbeil text)

- Any relationship between quantities that can be expressed using derivatives can also be expressed using integrals.
- Example: For position x(t), velocity v(t), and acceleration a(t),

- We saw in Week 6 that power is the derivative with respect to time of energy:
- Therefore energy is the integral with respect to time of power (plus the initial energy):

- We saw in Week 6 that, for a capacitor,
- Therefore, for a capacitor,

- We saw in Week 6 that, for an inductor,
- Therefore, for an inductor,

- Current, voltage & energy in a capacitor
- Current & voltage in an inductor (graphical)
- Current & voltage in a capacitor (graphical)
- Current & voltage in a capacitor (graphical)

- Recall that:
- Differentiating a parabola gives a slant line.
- Differentiating a slant line gives a horizontal line (constant).
- Differentiating a horizontal line (constant) gives zero.

- Therefore:
- Integrating zero gives a horizontal line (constant).
- Integrating a horizontal line (constant) gives a slant line.
- Integrating a slant line gives a parabola.

- The change in position between times t1 and t2 is equal to the area under the velocity curve between t1 and t2:

- For an inductor, the change in current between times t1 and t2 is equal to 1/L times the area under the voltage curve between t1 and t2:

- For a capacitor, the change in voltage between times t1 and t2 is equal to 1/C times the area under the current curve between t1 and t2: