Area Region R is bounded by the curves y 2 x 2 and y x Sketch region R R What is the area of region R Process To find the area between curves Sketch the region defined in the problem ID: 254241
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Slide1
7.2 Areas Between CurvesSlide2
Area
Region
R is bounded by the curves y = 2 – x2 and y = -x.Sketch region R.
R
What is the area of region
R
?Slide3
Process
To find the area between curves:
Sketch the region defined in the problem.Connect the curves with either a vertical strip (dx) or a horizontal strip (dy).A strip that always connects the two curves will allow you to find the area without breaking up integrals.
Write an expression for the length of the rectangular strips.
Vertical Strips: Length = Top curve – Bottom curveHorizontal Strips: Length = Right curve – Left curveNOTE:
IF YOU USE A dy STRIP, YOU MUST SOLVE THE CURVE FOR x IN TERMS OF y.Add rectangular strips together by setting up an integral using your expression.
Find points of intersection. NOTE: If using a dx, use the x-coordinates of intersection. If using a dy
, use the y-coordinates of intersection.Slide4
Area
What is the area of region
R
?
dx
Using a
dx
strip because it always connects the two curves.
Length of
dx
strip = top – bottom
y = 2 – x
2
y = -x
= (2 – x
2
) – (-x)
= –x
2
+ x + 2
Intersection
Bounds???
2 – x
2
= –x
–x
2
+ x + 2 = 0
–1(x + 1)(x – 2) = 0
–1(x
2
–
x – 2) = 0
x = –1 x = 2Slide5Slide6
Example
Find the area bounded by
y = ex, y = e2
, and the y-axis.
Strip?
dx
Length?
Bounds?
x = 0
and intersection (
e
2
= e
x
x = 2)Slide7
Example
Find the area between the two curves
x = y2 – 4y and y = x bounded by the x-axis.
Strip?
dy
Length?
Right – Left
Bounds?
y = 0
and intersection (
y
2
– 4y = y
y = 5
)Slide8
Homework
Section 7.2 (#1-25 odd, 27-42 multiples of 3, 48)