Sections 4.2-4.4: Area, Definite Integrals, and The Fundamental Theorem of Calculus
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Transcript of Sections 4.2-4.4: Area, Definite Integrals, and The Fundamental Theorem of Calculus
Distance TraveledA train moves along a track at a steady rate of 75 miles per
hour from 3:00 A.M. to 8:00 A.M. What is the total distance traveled by the train?
Time (h)
Velo
city
(mph
) Applying the well known formula:
distance rate time 75mph 5hr 375 miles
Notice the distance traveled by the train (375 miles) is exactly the area of the
rectangle whose base is the time interval [3,8] and whose height is the constant
velocity function v=75.
Distance TraveledA particle moves along the x-axis with velocity v(t)=-t2+2t+5
for time t≥0 seconds. How far is the particle after 3 seconds?
Time
Velo
city
The distance traveled is still the area under the curve.
Unfortunately the shape is a irregular region. We need to
find a method to find this area.
The Area ProblemWe now investigate how to solve the area
problem: Find the area of the region S that lies under the curve y=f(x) from a to b.
a b
f(x)
S
This means S is bounded by the graph of a continuous function, two vertical lines, and
the x-axis.
Finding AreaIt is easy to calculate the area of certain
shapes because familiar formulas exist:
A=lw A=½bhThe area of irregular polygons can be found
by dividing them into convenient shapes and their areas:
A1
A2 A
3 A4
1 2 3 4A A A A A
Approximating the Area Under a Curve
a b
We first approximate the area under a function by rectangles.
Approximating the Area Under a Curve
a b
Then we take the limit of the areas of these rectangles as we increase the number of rectangles.
Approximating the Area Under a Curve
a b
Then we take the limit of the areas of these rectangles as we increase the number of rectangles.
Estimating Area Using Rectangles and Right Endpoints
Use rectangles to estimate the area under the parabola y=x2 from 0 to 1 using 4 rectangles and right endpoints.
12
14
34
Divide the area under the curve into 4 equal strips
Make rectangles whose base is the same as the strip and whose height is the same as
the right edge of the strip.
Find the Sum of the Areas:
1 14 4A f 1 1
4 2f 314 4f 1
4 1f
21 14 4 21 1
4 2 2314 4 21
4 1
0.46875
Width = ¼ and height = value of the function
at ¼
Estimating Area Using Rectangles and Right Endpoints
Use rectangles to estimate the area under the parabola y=x2 from 0 to 1 using 8 rectangles and right endpoints.
12
14
34
Divide the area under the curve into 8 equal strips
Make rectangles whose base is the same as the strip and whose height is the same as
the right edge of the strip.
Find the Sum of the Areas:
1 18 8A f 1 1
8 4f 318 8f 1 1
8 2f
0.3984375
Width = 1/8 and height = value of the
function at 1/8
38
58
78
18
518 8f 31
8 4f 718 8f 1
8 1f
21 18 8 21 1
8 4 2318 8 21 1
8 2
2518 8 231
8 4 2718 8 21
8 1
Estimating Area Using Rectangles and Left Endpoints
Use rectangles to estimate the area under the parabola y=x2 from 0 to 1 using 4 rectangles and left endpoints.
12
14
34
Divide the area under the curve into 4 equal strips
Make rectangles whose base is the same as the strip and whose height is the same as
the left edge of the strip.
Find the Sum of the Areas:
14 0A f 1 1
4 4f 1 14 2f 31
4 4f
214 0 21 1
4 4 21 14 2 231
4 4
0.21875
Width = ¼ and height = value of the function
at 0
Estimating Area Using Rectangles and Left Endpoints
Use rectangles to estimate the area under the parabola y=x2 from 0 to 1 using 8 rectangles and left endpoints.
12
14
34
Divide the area under the curve into 8 equal strips
Make rectangles whose base is the same as the strip and whose height is the same as
the left edge of the strip.
Find the Sum of the Areas:
18 0A f 1 1
8 8f 1 18 4f 31
8 8f
0.2734375
Width = 1/8 and height = value of the
function at 0
38
58
78
18
1 18 2f 51
8 8f 318 4f 71
8 8f
218 0 21 1
8 8 21 18 4 231
8 8
21 18 2 251
8 8 2318 4 271
8 8
Distance TraveledA particle moves along the x-axis with velocity v(t)=-t2+2t+5
for time t≥0 seconds. Use three midpoint rectangles to estimate how far the particle traveled after 3 seconds?
Time
Velo
city
Divide the area under the curve into 3 equal strips
Make rectangles whose base is the same as the strip and whose height is the same as
the middle of the strip.
Find the Sum of the Areas:
1 0.5D v 1 1.5v 1 2.5v
21 0.5 2 0.5 5
21 1.5 2 1.5 5
21 2.5 2 2.5 5
15.25
Width = 1 and height = value of the function at
0.5
Negative AreaIf a function is less than zero for an interval,
the region between the graph and the x-axis represents negative area.
Positive Area
Negative Area
Definite Integral: Area Under a Curve
If y=f(x) is integrable over a closed interval [a,b], then the area under the curve y=f(x) from a to b is the integral of f from a to b.
area above the -axis area below the -axisb
af x dx x x
Upper limit of integration
Lower limit of integration
The Existence of Definite IntegralsAll continuous functions are integrable. That is, if a function
f is continuous on an interval [a,b], then its definite integral over [a,b] exists .
Ex: 9
14 x dx 1
2 3 3 12 5 5
8
Rules for Definite Integrals
Constant Multiple
Sum Rule
Difference Rule
Additivity
a a
b bkf x dx k f x dx
a a a
b b bf x g x dx f x dx g x dx
a a a
b b bf x g x dx f x dx g x dx
Let f and g be functions and x a variable; a, b, c, and k be constant.
b c c
a b af x dx f x dx f x dx
The First Fundamental Theorem of Calculus
If f is continuous on the interval [a,b] and F is any function that satisfies F '(x) = f(x) throughout this interval then
b
af x dx F b F a
b
av t dt s b s a
Example 1Evaluate
First Find the indefinte integral
F(x): 2F x x dx 2 11
2 1 x C 31
3 x C
13
Now apply the FTC to find the definite
integral:
F b F a 1 0F F
3 31 13 31 0C C
13 C C Notice that it is
not necessary to include the “C”
with definite integrals
1 2
0x dx