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Warm-Up: January 31, 2013. 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. Homework Questions?. Definite Integrals. Section 5.2. Sigma Notation. The Greek capital Sigma, Σ , stands for “sum.”

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Warm up january 31 2013
Warm-Up: January 31, 2013

  • 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

Definite Integrals

Section 5.2


Sigma notation
Sigma Notation

  • The Greek capital Sigma, Σ, stands for “sum.”

  • The index, k, tells us where to start (below Σ) and where to end (above Σ).


Riemann sums
Riemann Sums

  • 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


Riemann sums1
Riemann Sums

  • 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


Riemann sums2
Riemann Sums

  • 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


Riemann sums3
Riemann Sums

  • 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]


Question
Question

  • Have we done any calculus yet?


The definite integral
The Definite Integral

  • I is the definite integral of f over [a, b]

  • ||P|| is the longest subinterval length, called the norm of the partition


Definite integrals of continuous functions
Definite Integrals ofContinuous Functions

  • 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



Integration notation1
Integration Notation

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


Assignment
Assignment

  • Read Section 5.2 (pages 258-266)

  • Seriously, read Section 5.2


Warm up february 1 2013
Warm-Up: February 1, 2013

  • Use your RIEMANN program to estimate



Example 1
Example 1

  • Express the limit as a definite integral


You try 1
You-Try #1

  • Express the limit as a definite integral


Area under a curve
Area Under a Curve

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

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


Integrals and areas
Integrals and Areas

  • For any integrable function:


Evaluating integrals
Evaluating Integrals

  • We can use geometric areas to evaluate certain integrals, including:

    • Constant functions (rectangles)

    • Linear functions (trapezoid)

    • Semi-circles



Integrals of constants
Integrals of Constants

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



Integrals on ti 83
Integrals on TI-83

  • [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


You try 4
You-Try #4

  • Approximate the following to three decimal places:


Assignment1
Assignment

  • Read Section 5.2 (pages 258-266)

  • Page 267 Exercises 1-27 odd

  • Read Section 5.3 (pages 268-274)


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