Ch 5.2Graphical, Numerical, Algebraic by Finney Demana, Waits, Kennedy

Reimann SumsDuring the last class, we’ve looked at approximating the total distance traveled between times a and b, by summing up rectangles and either getting an overestimation or an underestimation.

LRAM RRAM

These approximations gotten from summing up a number of rectangles, are called Reimann sums.

Sigma Notation

n

n kk 1

S = f x k

The more rectangles you have within the same interval, the more accurate your approximation becomes. You have complete precision if you take the limit of these sums as the number of rectangles you use approaches infinity.

Definite Integral

Given that f(t) is non-negative, continuous for a ≤ t ≤ b. The definite integral of from a to b is written:

nb

ka nk 1

A = f t dt = lim f x k

Example

2 2

2Evaluate the integral 4 - x dx

2

1

-1

-2 2

Example

2 2

2Evaluate the integral 4 - x dx = 2

2

1

-1

-2 2

Notice that in this graph that both the width Δx and the height f(x) are positive which gives us positive area. What happens if f(x) is negative?

Example

Evaluate the integral sin x dx

1

-1

-2 2

Example

b

a

Evaluate the integral sin x dx = 0

Note : f x dx = area above axis - area below axis

1

-1

-2 2

Discontinuous Functions

Some functions with discontinuities are also integrable. A bounded function that has a finite number of points of discontinuity on an interval [a, b] will still be integrable on the interval if it is continuous everywhere else.

2

1

xFind dx

x

Discontinuous Function

2

1

x dx

x

= 1 1 + 1 2

= 1

2

1

-1

-2

-2 2

Try this on your calculator using fnInt…