Lesson 3 Oct 30
11
Definite Integrals
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Transcript of Lesson 3 Oct 30
Definite Integrals
limn
f(x1) x1 x2f(x2)+ xnf(xn)+ ..... +[ ]f(x) dx = a
b
a b
ff(x) dx =
a
bArea =
Given the integral (x + 1) dx = 1
4
Estimate the integral using left and right sums with 50 subdivisions.
left hand = 10.41
right hand = 10.59
average = 10.50
Given the integral (x + 1) dx = 1
4
Draw the graph and find the exact value of the area of the region.
Given f(x) = x
over the interval [2, 0].
approximate the definite integral f(x) dx = a
b
if a function is negativevalued, the definite integral is the negative of the area of the region between the graph of f and the xaxis
In General:
if a function is positivevalued, the definite integral
f(x) dx a
bis positive
(5 2x) dx = 0
4Evaluate:
Use graphing calculator to find integral value.
Error bounds for Monotone Functions
a function that is either increasing or decreasing but not both on an interval [a, b]
f
a b
f
a b
f(b)
f(a)
f(b) f(a)
Error difference between the lower and upper sum
= .f(b) f(a) b a n
( )
The only variable in this expression is the number of subintervals, n.
For the function f(x) = x2 on the interval [0,2], how many intervals of equal length must be used so that an estimate differs from the value of
x2 dx 0
2by less than 0.01
b a n
( )
f(b) f(a) 4 0
=
=
2n
4 0 . 2n( ) = 8
n
8n < 0.01
n > 800
