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Section 4.3 Riemann Sums and Definite Integrals

Section 4.3 Riemann Sums and Definite Integrals. Section 4.3 Riemann Sums and Definite Integrals.

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Section 4.3 Riemann Sums and Definite Integrals

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  1. Section 4.3 Riemann Sums and Definite Integrals

  2. Section 4.3 Riemann Sums and Definite Integrals • To this point, anytime that we have used the integral symbol we have used it without any upper or lower boundaries. In other words, it has served as simply a short hand symbol asking you to antidifferentiate a function. • We have also been talking about areas simultaneously. Those problems do have upper and lower bound limits. • There is a reason that we have been pursuing these two seemingly different goals at the same time.

  3. Section 4.3 Riemann Sums and Definite Integrals • The reason why is that they are the same goal, not different ones. • Using words to describe a problem: • Find the area of the region bounded by the curve ,the x-axis, and the lines x = 2 and x = 6. • Using summation to describe the problem • Using integral notation to describe the problem

  4. Section 4.3 Riemann Sums and Definite Integrals • Below is a visual representation of the region bounded by the curve and the vertical boundaries.

  5. Section 4.3 Riemann Sums and Definite Integrals • Draw the region defined by the integral below:

  6. Section 4.3 Riemann Sums and Definite Integrals • Did your graph look like the one below?

  7. Section 4.3 Riemann Sums and Definite Integrals • Now try to read a summation notation and turn it into a picture AND into a definite integral. Here’s the sigma notation:

  8. Section 4.3 Riemann Sums and Definite Integrals • Does your picture look like this?

  9. Section 4.3 Riemann Sums and Definite Integrals • Does your integral look like this?

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