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 = x 2 - 9, -6 x 2 - - - - - - - - f . I Sab f 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
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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