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# Warm-Up: January 31, 2013 - PowerPoint PPT Presentation

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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• 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

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

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)

• 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)