MATH 2414 Calculus II Notes
Section 6.2: Regions Between Curves
Christy Dittmar
Regions Between Curves [6.2]
Motivation
Velocities of two cyclists in a race are shown. Describe their relative positions at t = 3 hours,
assuming they have the same start position.
Area by Slicing
Definition: Area of a Region Between Two Curves
Suppose that f and g are continuous functions with f ( x ) ≥ g ( x ) on the interval [ a, b ] .
The area of the region bounded by the graphs of f and g on [ a, b ] is
=
A
∫ ( f ( x ) − g ( x ) ) dx
b
a
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MATH 2414 Calculus II Notes
Section 6.2: Regions Between Curves
Christy Dittmar
Integrating with respect to x
1. Give an exact value for the area of the region bounded by y = 2 and y = 2sin x , on the interval
 π
0, 2 
Integrating with respect to y
2. Determine the area of the shaded region. Give an exact value.
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MATH 2414 Calculus II Notes
Section 6.2: Regions Between Curves
Christy Dittmar
Variable of Integration
Slice ∆x
Slice ∆y
3. Determine the area of the shaded region. Give an exact value.
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MATH 2414 Calculus II Notes
Section 6.2: Regions Between Curves
Christy Dittmar
Activity
2
4. The region R is enclosed by the graphs of y = x , y= 6 − x , and y = 0 .
a. Draw a representative horizontal area slice (or slices) in the region.
i. Using this slice, what is your variable of integration ( x or y )?
ii. Can it be written as a single integral (yes or no)?
iii. Write the area integral(s) using horizontal slicing. Do not evaluate.
b. Draw a representative vertical area slice (or slices) in the region.
i. Using this slice, what is your variable of integration ( x or y )?
ii. Can it be written as a single integral (yes or no)?
iii. Write the area integral(s) using vertical slicing. Do not evaluate.
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MATH 2414 Calculus II Notes
Section 6.2: Regions Between Curves
Christy Dittmar
5. Find the exact and approximate area (to three decimal places) of the region shown.
Note: you can do two different u-substitutions by separating the integrals.
6. Use technology to check your answer. On www.desmos.com do the following:
a. Type f ( x ) =
16
in line 1 and g ( x )= 4 − 2 x in line 2
4 + x2
b. Click on the intersection points to find the coordinates.
c. Type Int to get a definite integral symbol in line 3, then turn it into ∫ ( f ( x ) − g ( x ) ) dx
2
0
d. To see the shaded region, type g ( x ) ≤ y ≤ f ( x ) in line 4 (type <= to get an inclusive inequality)
See also: Calculus II Technology Summary
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MATH 2414 Calculus II Notes
Section 6.2: Regions Between Curves
Christy Dittmar
Extra Practice
7. Find the exact area of the region bounded by y = e x =
, y 6e − x + 5 , and x = 0 .
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