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Warm-Up: January 31, 2013

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- Use your RIEMANN program to find the area under f(x) from x=a to x=b using each rectangular approximation method (LRAM, MRAM, and RRAM) with n=10

Definite Integrals

Section 5.2

- The Greek capital Sigma, Σ, stands for “sum.”
- The index, k, tells us where to start (below Σ) and where to end (above Σ).

- RRAM, MRAM, and LRAM are examples of Riemann sums
- Function f(x) on interval [a, b]
- We partitioned the interval into nsubintervals (to make the bases of our n rectangles).
- Let’s name the x-coordinates at the corners of our rectangles

- Let’s denote a by x0 and b by xn
- The corners of our rectangles make the set
- P is called a partition of [a,b]
- Our subintervals are the closed intervals
- The width of the kth subinterval has width

- We choose some x value inside each subinterval to be the height of the rectangle.
- RRAM: Use the right endpoint
- MRAM: Use the midpoint
- LRAM: Use the left endpoint
- Could use any other point

- Let’s call the x value from the kth subinterval ck

- The area of each rectangle is the product of its height, f(ck), and its width, Δxk
- The sum of these areas is
- This is the Riemann Sum for f on the interval [a, b]

- Have we done any calculus yet?

- I is the definite integral of f over [a, b]
- ||P|| is the longest subinterval length, called the norm of the partition

- All continuous functions are integrable.
- Let f be continuous on [a, b]
- Let [a, b] be partitioned into n subintervals of equal length
- The definite integral is given by
- where each ck is chosen arbitrarily in the kth subinterval

The function is the integrand

Upper limit of integration

x is the variable of integration

Integral sign

Lower limit of integration

Integral of f from a to b

- Read Section 5.2 (pages 258-266)
- Seriously, read Section 5.2

- Use your RIEMANN program to estimate

- Express the limit as a definite integral

- Express the limit as a definite integral

- If y=f(x) is nonnegative and integrable on [a, b]
- If y=f(x) is nonpositive and integrable on [a, b]

- For any integrable function:

- We can use geometric areas to evaluate certain integrals, including:
- Constant functions (rectangles)
- Linear functions (trapezoid)
- Semi-circles

- If f(x)=c, where c is a constant, then:

- [MATH] [9:fnInt]
- fnInt(function, X, lower bound, upper bound)
- or
- Enter function into Y1
- [2nd] [TRACE] [7:∫f(x)dx]
- Enter lower limit and upper limit

- Approximate the following to three decimal places:

- Read Section 5.2 (pages 258-266)
- Page 267 Exercises 1-27 odd
- Read Section 5.3 (pages 268-274)