Therefore, your average test grade is approximately 80.33, which translates to a B− at most schools.
Suppose, however, that we have a function
that gives us the speed of an object at any time
t , and we want to find the object’s average speed. The function
takes on an infinite number of values, so we can’t use the process just described. Fortunately, we can use a definite integral to find the average value of a function such as this.
Let
be continuous over the interval
and let
be divided into
n subintervals of width
Choose a representative
in each subinterval and calculate
for
In other words, consider each
as a sampling of the function over each subinterval. The average value of the function may then be approximated as
which is basically the same expression used to calculate the average of discrete values.
But we know
so
and we get
Following through with the algebra, the numerator is a sum that is represented as
and we are dividing by a fraction. To divide by a fraction, invert the denominator and multiply. Thus, an approximate value for the average value of the function is given by
This is a Riemann sum. Then, to get the
exact average value, take the limit as
n goes to infinity. Thus, the average value of a function is given by
Definition
Let
be continuous over the interval
Then, the
average value of the function
(or
fave ) on
is given by
Finding the average value of a linear function
Find the average value of
over the interval
First, graph the function on the stated interval, as shown in
[link] .
The region is a trapezoid lying on its side, so we can use the area formula for a trapezoid
where
h represents height, and
a and
b represent the two parallel sides. Then,
The definite integral can be used to calculate net signed area, which is the area above the
x -axis less the area below the
x -axis. Net signed area can be positive, negative, or zero.
The component parts of the definite integral are the integrand, the variable of integration, and the limits of integration.
Continuous functions on a closed interval are integrable. Functions that are not continuous may still be integrable, depending on the nature of the discontinuities.
The properties of definite integrals can be used to evaluate integrals.
The area under the curve of many functions can be calculated using geometric formulas.
The average value of a function can be calculated using definite integrals.
Key equations
Definite Integral
Properties of the Definite Integral
for constant
c
Questions & Answers
Discuss the differences between taste and flavor, including how other sensory inputs contribute to our perception of flavor.
The lymphatic system plays several crucial roles in the human body, functioning as a key component of the immune system and contributing to the maintenance of fluid balance. Its main functions include:
1. Immune Response: The lymphatic system produces and transports lymphocytes, which are a type of
asegid
to transport fluids fats proteins and lymphocytes to the blood stream as lymph
Anatomy is the study of the structure of the body, while physiology is the study of the function of the body. Anatomy looks at the body's organs and systems, while physiology looks at how those organs and systems work together to keep the body functioning.
Enzymes are proteins that help speed up chemical reactions in our bodies. Enzymes are essential for digestion, liver function and much more. Too much or too little of a certain enzyme can cause health problems
Kamara
yes
Prince
how does the stomach protect itself from the damaging effects of HCl
the normal temperature is 37°c or 98.6 °Fahrenheit is important for maintaining the homeostasis in the body
the body regular this temperature through the process called thermoregulation which involves brain skin muscle and other organ working together to maintain stable internal temperature