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

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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

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Area Between Two CurvesThe 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 ChallengeFind 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

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Between

Outside

4

In this example, all of the area was above the x-axis.

Does the same process work for “negative” area?

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Subtracting the bottom area from the top, leaves

only the area in-between.

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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 dxBetween

(Positive)

Between(Negative)

b

af x g x dx

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In this example, one area was positive and one was negative. Does the same process work if both areas

are negative?

THE SAME!

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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

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In this example, both areas were negative. Now we can apply the

three scenarios to any two curves.

(Counted Twice)

THE SAME!

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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

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POS-POS POS-NEG

NEG-NEG

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Area Between Two CurvesIf 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

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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

163

Find the Boundaries/Intersections1,3x

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 13

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 CurvesIf 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.

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“Warm-up”: 1985 Section I

NOW WE CAN DO!

1 3 140

8 8x x dx

White Board ChallengeFind 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

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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