18Section Lecture 5.5 Some Area Problems
Jiwen He
Quiz 1 What is today? a. b. c. d. Monday Wednesday Friday None of these
1
1.1
Section 5.5 Some Area Problems
Area below the graph of a Nonnegative f
Area below the graph of a Nonnegative f f (x) 0 for all x in [a, b].
= region below the graph of f .
1
b
Area of =
a
f (x) dx = F (b) F (a)
where F (x) is an antiderivative of f (x). Fundamental Theorem of Integral Calculus Theorem 1. In general,
b
f (x) dx = F (b) F (a).
a
where F (x) is an antiderivative of f (x).
2
Quiz 2
1
Give the value of
1
x3 2x2 + sin(x) dx. 1 2 4 3 4 3 1 2
a. b. c. d. e.
None of these
Example 1 Example 2. Find the area below the graph of the square-root function from x = 0 to x = 1.
3
Example 2 Example 3. Find the area bounded above by the curve y = 4 x2 and below by the x-axis.
Quiz 3 Give the area bounded between the x-axis and the graph of y = x2 + 1 for
4
1 x 2. a. b. c. d. e. 5 4 3 2 None of these
1.2
Area between the graphs of f and g
Area between the graphs of two Nonnegative f and g
f (x) g(x) 0
for all x in [a, b].
= region between the graphs of f (Top) and g (Bottom).
b b
Area of =
a
Top Bottom dx =
a
f (x) g(x) dx.
Example 3 Example 4. Find the area bounded above by y = x + 2 and below by y = x2 .
5
Area between the graphs of f and g
f (x) g(x)
for all x in [a, b].
= region between the graphs of f (Top) and g (Bottom).
b b
Area of =
a
Top Bottom dx =
a
f (x) g(x) dx.
Example 4 Example 5. Find the area of the region shown in the gure below.
6
Example 5 Example 6. Find the area between y = 4x and y = x3 from x = 2 to x = 2.
Example 6 Example 7. Use integrals to represent the area of the region = 1 2 shaded in the gure below.
7
1.3
c a
Signed Area
f (x) as dx Signed Area f (x) 0
b
for all x in [a, b]
f (x) dx = Area of 1
a
f (x) 0
c b
for all x in [b, c]
f (x) dx = Area of 2
8
c
b
c
f (x) dx =
a a
f (x) dx +
b
f (x) dx = Area of 1 Area of 2
= Area above the x-axis Area below the x-axis.
b a
f (x) dx as Signed Area
9
b
c
d
e
b
f (x) dx =
a a
f (x) dx +
c
f (x) dx +
d
f (x) dx +
e
f (x) dx
= Area of 1 Area of 2 + Area of 3 Area of 4 = Area of 1 + Area of 3 Area of 2 + Area of 4 = Area above the x-axis Area below the x-axis.
Example 7 Example 8. Evaluate
3
x2 2x dx and interpret the result in terms of areas.
1
Example 8 Example 9. Use integrals to represent the area of the region shaded in the gure below.
10
Quiz 4 The graph of y = f (x) is shown below. 1 has area 4 , 2 has area 4 , and 3 3
3 4 3 has area 3 . Give
f (x) dx.
1
a. b. c. d. e.
0 4 3 8 3 4 None of these
Quiz 5 4 The graph of y = f (x) is shown below. 1 has area 3 , 2 has area 4 , and 3
2 4 3 has area 3 . Give
f (x) dx.
1
11
a. b. c. d. e.
0 4 3 8 3 4 None of these
Quiz 6 4 The graph of y ...