section 7.1 – area of a region between two curves
TRANSCRIPT
Section 7.1 – Area of a Region Between Two Curves
White Board Challenge
The circle below is inscribed into a square:
What is the shaded area?
20 cm
2400 100 85.841 cm
Calcu
lator
White Board ChallengeFind the area of the region bounded by the function
below and the x-axis between x = 1 to x = 6:
6
1
6 2
10.1 5 2
12.16
f x dx
x dx
20.1 5 2f x x
Calcu
lator
Area Between Two Curves
The area of a region that is bounded above by one curve, y = f(x), and below by another y = g(x).
The area is always
POSITIVE.
White Board Challenge
Find the area of the region between y = sec2x and y = sin x from x = 0 to x = π/4:
4 2
0sec x dx
2secy x
4
0sin x dx
siny x
Area between the curves
4 2
0sec sinx x dx
22
TOP BOTTOM
Between
Outside
4
In this example, all of the area was above the x-axis.
Does the same process work for “negative” area?
Calcu
lator
Subtracting the bottom area from the top, leaves
only the area in-between.
TOP BOTTOM
Area Between Two Curves: Positive and Negative Area
Find the area of the region between the two curves from x = a to x = b:
g x
f x
ba
Area between the curves
b
af x dx
b
ag x dx
Between(Positive)
Between(Negative)
b
af x g x dx
TOP BOTTOM
In this example, one area was positive and one was negative. Does the same process work if both areas
are negative?
THE SAME!
TOP BOTTOM
Subtracting the negative area switches it to adding
a positive version.
Must be positive!
Area Between Two Curves: Negative Area Only
Find the area of the region between the two curves from x = a to x = b:
g x
f x
ba Area between the curves
b
af x dx
b
ag x dx
Outside
Between (Negative)
b
af x g x dx
TOP BOTTOM
In this example, both areas were negative. Now we can apply the
three scenarios to any two curves.
(Counted Twice)
THE SAME!
TOP BOTTOM
Subtracting the negative area switches it to adding
a positive version AND cancels the outside area.
Area Between Two Curves: A Mix
Find the area of the region between the two curves from x = a to x = b:
g x
f x
ba
Area between the curves
b
af x dx
b
ag x dx
b
af x g x dx
TOP BOTTOM
POS-POS POS-NEG
NEG-NEG
TOP BOTTOM
Area Between Two Curves
If f and g are continuous functions on the interval [a,b], and if f(x) ≥ g(x) for all x in [a,b], then the area of the region bounded above by y = f(x), below by y = g(x), on the left by x = a, and on the right by x = b is:
b
aA f x g x dx
TOP BOTTOM
Reminder: Riemann Sums Recall that the integral is a limit of Riemann Sums:
f x
g x
* *k kf x g x
kxArea
* *k k kf x g x x
1
n
kmax 0
limkx
b
af x g x dx a b
Example 1Find the area of the region between the graphs of the functions
2 24 10, 4 , 1 3f x x x g x x x x Sketch a Graph
Make Generic “Riemann”
Rectangle(s)
Base = dx
Height = f – g
Integrate the Area of Each Generic Rectangle
3 2 2
14 10 4x x x x dx
16
3
Find the Boundaries/Intersections
1,3x
Example 2Find the area of the region enclosed by the parabolas y = x2 and y = 2x – x2.
Sketch a Graph
Make Generic “Riemann”
Rectangle(s)
Base = dx
Height = (2x–x2)–(x2)
Integrate the Area of Each Generic Rectangle
1 2 2
02x x x dx 1
3
Find the Boundaries/Intersections2 22x x x
0,1x
22 2 0x x 2 1 0x x
Example 3Find the area of the region bounded by the graphs y = 8/x2, y = 8x, and y = x.
Sketch a Graph
Make Generic “Riemann” Rectangle(s)
Base = dx
Height = 8x-x
Integrate the Area of Each Generic Rectangle
2
1 28
0 18
xx x dx x dx
6
Find the Boundaries/Intersections2
8x
x 2x
Base = dx
Height = 8/x2-x
8x x0x
288x
x 1x
2
Example 4Find the area of the region bounded by the curves y = sin x, y = cos x, x = 0, and x = π/2.
Sketch a Graph
Make Generic “Riemann”
Rectangle(s)
Base = dx
Height = cos-sin
Integrate the Area of Each Generic Rectangle
4 2
0 4cos sin sin cosx x dx x x dx
2 2 2
Find the Boundaries/Intersectionssin cosx x
4x Base = dx
Height = sin-cos
20,x
What other Integrals could be used?
4
02 cos sinx x dx
2
0cos sinx x dx
(Symmetrical)
(Keeps it Positive)
Area Between Two Curves
If f and g are continuous functions on the interval [a,b], then the area of the region bounded by y = f(x), y = g(x), on the left by x = a, and on the right by x = b is:
b
aA f x g x dx
It does not matter which function is greater.
NOTE: There have been AP problems in the past that ask for an integral without an absolute value. So the first method is still
important.
No Calculator
“Warm-up”: 1985 Section I
NOW WE CAN DO!
1 3 140
8 8x x dx
White Board Challenge
Find the area enclosed by the line y = x – 1 and the parabola y2 = 2x + 6.
1
3
1
3
2 6 2 6
2 6 1
18
x x dx
x x dx
2 6y x
2 6y x
1y x
Calcu
lator
Example 5Find the area enclosed by the line y = x – 1 and the
parabola y2 = 2x + 6. Sketch a Graph
Make Generic “Riemann” Rectangle(s)
Base = dy
Height=(y+1)–(1/2y2–3)
Integrate the Area of Each Generic Rectangle
4 2122
1 3y y dy
18
Find the Boundaries/Intersections21
21 3y y
2,4y
20 2 8y y 0 4 2y y
Sometimes Solve for x1y x 1x y
2 2 6y x 21
2 3x y
White Board ChallengeUsing two methods (one with dx and one with dy), find
the area between the x-axis and the two curves:
2 4
0 2
103
2x dx x x dx
2y x or x y
2 2y x or x y
& 2y x y x
2 2 1030
2y y d
OR
y