mat 212 brief calculus section 5.4 the definite integral
TRANSCRIPT
MAT 212Brief Calculus
Section 5.4The Definite Integral
The Definite Integral
• Given a function f that is continuous on the interval [a,b], if we divide the interval into n subintervals and choose a point, in each interval, the definite integral from a to b is said to be
*ix
*
01
( ) lim ( )nb
ia ni
f x dx f x x
a and b are our limits of integration
The Definite Integral
• Given a function f that is continuous on the interval [a,b], if we divide the interval into n subintervals and choose a point, in each interval, the definite integral from a to b is said to be
*ix
*
01
( ) lim ( )nb
ia ni
f x dx f x x
f(x) is called our integrand
The Definite Integral
• Given a function f that is continuous on the interval [a,b], if we divide the interval into n subintervals and choose a point, in each interval, the definite integral from a to b is said to be
*ix
*
01
( ) lim ( )nb
ia ni
f x dx f x x
dx represents our rectangle widths and tells us what our variable of integration is
The Definite Integral
• Given a function f that is continuous on the interval [a,b], if we divide the interval into n subintervals and choose a point, in each interval, the definite integral from a to b is said to be
*ix
*
01
( ) lim ( )nb
ia ni
f x dx f x x
The definite integral calculates the net area underneath f(x) and above the x-axis between a and b
The Definite Integral
• Given a function f that is continuous on the interval [a,b], if we divide the interval into n subintervals and choose a point, in each interval, the definite integral from a to b is said to be
*ix
*
01
( ) lim ( )nb
ia ni
f x dx f x x
This can be thought of as the calculation of the accumulation of f
The Definite Integral
• Given a function f that is continuous on the interval [a,b], if we divide the interval into n subintervals and choose a point, in each interval, the definite integral from a to b is said to be
*ix
*
01
( ) lim ( )nb
ia ni
f x dx f x x
We saw that we can calculate a definite integral using the fundamental theorem of calculus, let’s take a quick look at the fundamental theorem
If f is continuous on the interval [a,b], and f(x) = F ‘(x), then
To find , first find F, then calculate F(b) - F(a)
This method of computing definite integrals gives an exact answer.
( ) ( ) ( )b
af x dx F b F a
( )b
af x dx
Examples
3 2
2
7
0
9
3
(2 4) ?
3 ?
1?
x dx
dx
dxx
Notation for the general antiderivative of a function f(x) looks like the definite integral
without the limits of integration.
( ) ( ) ( )b
af x dx F b F a
( ) ( )f x dx F x C
Definite:
General:
( ) ( ) ( )b
af x dx F b F a
( ) ( )f x dx F x C
Definite:
General:
This gives you a number
This represents a family of functions
The Definite Integral
If a, b, and c are any numbers and f is a continuous function, then
1. ( ) ( )
2. ( ) ( ) ( )
b a
a b
b c c
a b a
f x dx f x dx
f x dx f x dx f x dx
The Definite Integral
Let f and g be continuous functions and let c be a constant, then
1.__ ( ( ) ( )) ( ) ( )
2.__ ( ) ( )
b b b
a a a
b b
a a
f x g x dx f x dx g x dx
cf x dx c f x dx
The Definite Integral
Let f and g be continuous functions and let c be a constant, then
1. ( )
2. ( ) 0
b
a
a
a
cdx c b a
f x dx
Find the exact value of the area between by the graphs of
y = e x + 3 and y = x + 2 for 0 ≤ x ≤ 2
Find the exact value of the area between by the graphs of
y = x + 1 and y = 7 - x for 0 ≤ x ≤ 4
Find the exact value of the area enclosed by the graphs of
y = x2 and y = 2 - x2
Let’s look at #13 from the book