Area between two curves

Integration and Area Calculation

  • Introduction to Area Calculation

    • Procedure for setting up integrals to find the area under curves.

    • Using calculators to compute definite integrals post-setup.

Integral Setup Example

  • Example Regions

    • Set limits for area calculation.

    • Example discussed:

      • Lower limit: 0

      • Upper limit: 4

    • Guidance on using the calculator:

      • Access the integral menu: Second function then trace, select number seven (integral operator).

      • Input limits into the calculator directly.

Points of Intersection

  • Finding Intersection of Curves

    • Example given: Upper function is x + 2.

    • Steps to input limits, example used:

      • Lower limit: -1

      • Upper limit: 2

      • Calculate intersections using the graphing function.

Area Between Curves

  • Situation with Changing Upper Function

    • Two curves discussed:

      • Black Curve: x^3 - x^2

      • Gray Curve: y = 4x - x^2

    • Importance of finding points of intersection and accounting for changes in upper function across regions.

  • Split the region into two distinct parts based on intersections:

    • First integral setup for left region from -1 to 0.

    • Need to confirm which curve is upper in differing intervals.

Consumer Surplus

  • Definition and Example

    • Consumer surplus as the difference between what consumers are willing to pay versus the market price.

    • Example of purchasing an iPhone:

      • If bought at a lower price, the difference is the consumer surplus.

    • Use of integration to find consumer surplus under the demand curve.

Supply and Producer Surplus

  • Introduction to Producer Surplus

    • Defined as the area above the supply function and below the market price.

    • Example discussing a producer willing to sell for $5 while consumers willing to pay $10 captures the surplus.

Examination and Study Guidelines

  • Details about the upcoming final exam:

    • Date and time confirmed: Thursday next week at 12.

    • Cancelled study session for other obligations.

  • Assignment reminders and emails discussed:

    • Worksheet completion and communication for any clarifications.

    • Students encouraged to engage and ask questions.

Conclusion

  • Importance of setting up problems accurately and using calculators effectively for solutions.

  • Discussions on consumer and producer surplus emphasizing real-world applications of calculus.