The Area Between Two Curves Lesson 6.1. What If … ? We want to find the area between f(x) and g(x)...
-
Upload
georgiana-york -
Category
Documents
-
view
216 -
download
1
Transcript of The Area Between Two Curves Lesson 6.1. What If … ? We want to find the area between f(x) and g(x)...
When f(x) < 0
• Consider taking the definite integral for the function shown below.
• The integral gives a negative area (!?) We need to think of this in a different way
a b
f(x)
( )b
a
f x dx
Another Problem
• What about the area between the curve and the x-axis for y = x3
• What do you get forthe integral?
• Since this makes no sense – we need another way to look at it
23
2
x dx
Recall our look at odd functions on the interval [-a, a]
Solution
• We can use one of the properties of integrals
• We will integrate separately for -2 < x < 0 and 0 < x < 2
( ) ( ) ( )b c b
a a c
f x dx f x dx f x dx
2 0 23 3 3
2 2 0
x dx x dx x dx
We take the absolute value for the interval
which would give us a negative area.
We take the absolute value for the interval
which would give us a negative area.
General Solution
• When determining the area between a function and the x-axis Graph the function first Note the zeros of the function Split the function into portions
where f(x) > 0 and f(x) < 0 Where f(x) < 0, take
absolute value of the definite integral
Try This!
• Find the area between the function h(x)=x2 + x – 6 and the x-axis Note that we are not given the limits of
integration We must determine zeros
to find limits Also must take absolute
value of the integral sincespecified interval has f(x) < 0
Area Between Two Curves
• Consider the region betweenf(x) = x2 – 4 and g(x) = 8 – 2x2
• Must graph to determine limits
• Now consider function insideintegral Height of a slice is g(x) – f(x)
So the integral is 2
2
( ) ( )g x f x dx
The Area of a Shark Fin
• Consider the region enclosed by
• Again, we must split the region into two parts 0 < x < 1 and 1 < x < 9
( ) 9 9 ( ) 9f x x g x x x axis
Slicing the Shark the Other Way
• We could make these graphs as functions of y
• Now each slice isy by (k(y) – j(y))
( ) 9 9 ( ) 9f x x g x x x axis
2 21( ) 9 ( ) 9
9j y x y and k y x y
3
0
( ) ( )k y j y dy
Practice
• Determine the region bounded between the given curves
• Find the area of the region
2 6y x y x
Horizontal Slices
• Given these two equations, determine the area of the region bounded by the two curves Note they are x in terms of y
2
2
8x y
x y
Integration as an Accumulation Process
• Consider the area under the curve y = sin x
• Think of integrating as an accumulation of the areas of the rectangles from 0 to b
b
0
sinb
x dx
Integration as an Accumulation Process
• We can think of this as a function of b
• This gives us the accumulated area under the curve on the interval [0, b]
00
( ) sin cos ( ) cos 1b
bA b x dx x b
Applications
• The surface of a machine part is the region between the graphs of y1 = |x| and y2 = 0.08x2 +k
• Determine the value for k if the two functions are tangent to one another
• Find the area of the surface of the machine part