Section 6.6 Area Between Two Curves

Area Between Two Curves: The area A of the region bounded by the curves y = f(x) and

y = g(x) and the lines x = a and x = b, where f and g are continuous and f(x) � g(x) for all x in [a, b]

is given by the definite integral

A =

Z b

a

[f(x)� g(x)] dx

Area of a Curve under the x-Axis: If the graph of y = f(x) is below the x-axis on [a, b],

then the area A below the x-axis and above the graph of y = f(x) on [a, b] is

A = �Z b

a

f(x) dx

Steps for finding the Area Between a function f(x) and the x-axis on [a, b]:

Graph f(x) and find the x-intercept(s) (You can do this on the calculator.)

Integrate the function on the interval [a, b], breaking it up around any x-intercepts, remembering

to multiply the integral by �1 if the function is below the x-axis.

1. Determine the area that is bounded by the following curve and the x-axis on the interval below.

(Round answer to three decimal places.)

y = x2 � 9, �6 x 2

--

-

- -

-

- -

f ..

I Sabf on d x I

-

-

-

-

f - 6,

2 3

x - i n t e r c e pts :

X ? 9 = O

( x - 3 X x t§ = O

⇒ x -

-- 3 r

3

i

'¥÷÷*v

a ÷÷÷÷÷::::÷÷÷.

= 36 t 3 3 a 333

= 69.333J

2. Determine the area that is bounded by the following curve and the x-axis on the interval below.

(Round answer to three decimal places.)

y = e�3x, �2 x 1

Steps for finding the Area Between two functions, f(x) and g(x), on [a, b]:

Graph both f(x) and g(x) find the x-value(s) where f(x) and g(x) intersect. (You can do

this on the calculator.)

Integrate the di↵erence of the two functions on the interval [a, b], breaking it up around any

intersect values, remembering to use f(x)� g(x) if f(x) is on top, and g(x)� f(x) if g(x) is

on top.

3. Determine the area that is bounded by the graphs of the following equations.

y = 64x, y = x3

2 Summer 2019, Maya Johnson

ite i

,MATH

-29

^ ^

Area :S e-" dx

-2

(¥#¥>

=rs4

- -

64x=x3

⇒ 64 x - x3=0

÷÷÷÷÷÷:'

"

Area :

1%3.64×3*+1:*

. ⇒ ↳EMM

" ⇒

1024 t 1024 =2

4. Determine the area that is bounded by the graphs of the following equations. (Round answer to

three decimal places.)

y = 3x, y = 9x� x2

5. Determine the area that is bounded by the graphs of the following equations on the interval below.

(Round answer to three decimal places.)

y = x2 + 7x, y = 8x+ 56, �2 x 2

3 Summer 2019, Maya Johnson

3 x = 9 x - x'

÷÷÷÷÷÷÷.

Area : I l 3 x - L 9 x - xD I dx I

= I - 361 ' ③

X2

t 7 × = 8 x t 56

XZ t 7 x - 8 x - 5 6 = O

±÷÷÷÷÷.

Area : I x2- x - 562 d x I = I - 562 .

5 I

= 562.5J

6. Determine the area that is bounded by the graphs of the following equations. (Round answer to

three decimal places.)

y = �x2, y = x3 � 6x

7. The graph of f is shown. Evaluate each integral by interpreting it in terms of areas.

(a)

Z 28

20

f(x) dx

(b)

Z 36

0

f(x) dx

4 Summer 2019, Maya Johnson

^

X 3- 6 x = - X2

--

xx

:÷±:::

If,

""

X ( x t 3)( x -2) = O

⇒ X = - 3,

O ,2

°

Area:/flyI 6×tx7d×/ t / §x3 - 6xtx7dx/

- 3

= I 15.751 t I - 5 .3331 I I 5.75 t 5. 333221.083J

-

¥i÷.

= I (8×-12) = -480

*MB°

= (8×4)+1218×8 ) t 14h12 ) t It 8) ( 12 )

± 1600