Sections 4.2-4.4: Area, Definite Integrals, and The Fundamental
Theorem of Calculus
Slide 2
Distance Traveled A 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) Velocity (mph) Applying
the well known formula: 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.
Slide 3
Distance Traveled A particle moves along the x -axis with
velocity v(t)=-t 2 +2t+5 for time t0 seconds. How far is the
particle after 3 seconds? Time Velocity 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.
Slide 4
The Area Problem We 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. ab f(x)f(x) S This means S is bounded by the
graph of a continuous function, two vertical lines, and the
x-axis.
Slide 5
Finding Area It is easy to calculate the area of certain shapes
because familiar formulas exist: A=lw A=bh The area of irregular
polygons can be found by dividing them into convenient shapes and
their areas: A1A1 A2A2 A3A3 A4A4
Slide 6
Approximating the Area Under a Curve a b We first approximate
the area under a function by rectangles.
Slide 7
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.
Slide 8
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.
Slide 9
Estimating Area Using Rectangles and Right Endpoints Use
rectangles to estimate the area under the parabola y=x 2 from 0 to
1 using 4 rectangles and right endpoints. 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: Width = and height = value of the
function at
Slide 10
Estimating Area Using Rectangles and Right Endpoints Use
rectangles to estimate the area under the parabola y=x 2 from 0 to
1 using 8 rectangles and right endpoints. 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: Width = 1/8 and height = value of
the function at 1/8
Slide 11
Estimating Area Using Rectangles and Left Endpoints Use
rectangles to estimate the area under the parabola y=x 2 from 0 to
1 using 4 rectangles and left endpoints. 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: Width = and height = value of the
function at 0
Slide 12
Estimating Area Using Rectangles and Left Endpoints Use
rectangles to estimate the area under the parabola y=x 2 from 0 to
1 using 8 rectangles and left endpoints. 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: Width = 1/8 and height = value of
the function at 0
Slide 13
Distance Traveled A particle moves along the x -axis with
velocity v(t)=-t 2 +2t+5 for time t0 seconds. Use three midpoint
rectangles to estimate how far the particle traveled after 3
seconds? Time Velocity 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: Width = 1 and height = value of the function at
0.5
Slide 14
Negative Area If 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
Slide 15
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. Upper limit of
integration Lower limit of integration
Slide 16
The Existence of Definite Integrals All 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:
Slide 17
Rules for Definite Integrals Constant Multiple Sum Rule
Difference Rule Additivity Let f and g be functions and x a
variable; a, b, c, and k be constant.
Slide 18
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
Slide 19
Example 1 Evaluate First Find the indefinte integral F(x): Now
apply the FTC to find the definite integral: Notice that it is not
necessary to include the C with definite integrals
