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Using the properties of the definite integral

Use the properties of the definite integral to express the definite integral of f ( x ) = −3 x 3 + 2 x + 2 over the interval [ −2 , 1 ] as the sum of three definite integrals.

Using integral notation, we have −2 1 ( −3 x 3 + 2 x + 2 ) d x . We apply properties 3. and 5. to get

−2 1 ( −3 x 3 + 2 x + 2 ) d x = −2 1 −3 x 3 d x + −2 1 2 x d x + −2 1 2 d x = −3 −2 1 x 3 d x + 2 −2 1 x d x + −2 1 2 d x .
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Use the properties of the definite integral to express the definite integral of f ( x ) = 6 x 3 4 x 2 + 2 x 3 over the interval [ 1 , 3 ] as the sum of four definite integrals.

6 1 3 x 3 d x 4 1 3 x 2 d x + 2 1 3 x d x 1 3 3 d x

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Using the properties of the definite integral

If it is known that 0 8 f ( x ) d x = 10 and 0 5 f ( x ) d x = 5 , find the value of 5 8 f ( x ) d x .

By property 6.,

a b f ( x ) d x = a c f ( x ) d x + c b f ( x ) d x .

Thus,

0 8 f ( x ) d x = 0 5 f ( x ) d x + 5 8 f ( x ) d x 10 = 5 + 5 8 f ( x ) d x 5 = 5 8 f ( x ) d x .
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If it is known that 1 5 f ( x ) d x = −3 and 2 5 f ( x ) d x = 4 , find the value of 1 2 f ( x ) d x .

−7

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Comparison properties of integrals

A picture can sometimes tell us more about a function than the results of computations. Comparing functions by their graphs as well as by their algebraic expressions can often give new insight into the process of integration. Intuitively, we might say that if a function f ( x ) is above another function g ( x ) , then the area between f ( x ) and the x -axis is greater than the area between g ( x ) and the x -axis. This is true depending on the interval over which the comparison is made. The properties of definite integrals are valid whether a < b , a = b , or a > b . The following properties, however, concern only the case a b , and are used when we want to compare the sizes of integrals.

Comparison theorem

  1. If f ( x ) 0 for a x b , then
    a b f ( x ) d x 0 .
  2. If f ( x ) g ( x ) for a x b , then
    a b f ( x ) d x a b g ( x ) d x .
  3. If m and M are constants such that m f ( x ) M for a x b , then
    m ( b a ) a b f ( x ) d x M ( b a ) .

Comparing two functions over a given interval

Compare f ( x ) = 1 + x 2 and g ( x ) = 1 + x over the interval [ 0 , 1 ] .

Graphing these functions is necessary to understand how they compare over the interval [ 0 , 1 ] . Initially, when graphed on a graphing calculator, f ( x ) appears to be above g ( x ) everywhere. However, on the interval [ 0 , 1 ] , the graphs appear to be on top of each other. We need to zoom in to see that, on the interval [ 0 , 1 ] , g ( x ) is above f ( x ) . The two functions intersect at x = 0 and x = 1 ( [link] ).

A graph of the function f(x) = sqrt(1 + x^2) in red and g(x) = sqrt(1 + x) in blue over [-2, 3]. The function f(x) appears above g(x) except over the interval [0,1]. A second, zoomed-in graph shows this interval more clearly.
(a) The function f ( x ) appears above the function g ( x ) except over the interval [ 0 , 1 ] (b) Viewing the same graph with a greater zoom shows this more clearly.

We can see from the graph that over the interval [ 0 , 1 ] , g ( x ) f ( x ) . Comparing the integrals over the specified interval [ 0 , 1 ] , we also see that 0 1 g ( x ) d x 0 1 f ( x ) d x ( [link] ). The thin, red-shaded area shows just how much difference there is between these two integrals over the interval [ 0 , 1 ] .

A graph showing the functions f(x) = sqrt(1 + x^2) and g(x) = sqrt(1 + x) over [-3, 3]. The area under g(x) in quadrant one over [0,1] is shaded. The area under g(x) and f(x) is included in this shaded area. The second, zoomed-in graph shows more clearly that equality between the functions only holds at the endpoints.
(a) The graph shows that over the interval [ 0 , 1 ] , g ( x ) f ( x ) , where equality holds only at the endpoints of the interval. (b) Viewing the same graph with a greater zoom shows this more clearly.
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Average value of a function

We often need to find the average of a set of numbers, such as an average test grade. Suppose you received the following test scores in your algebra class: 89, 90, 56, 78, 100, and 69. Your semester grade is your average of test scores and you want to know what grade to expect. We can find the average by adding all the scores and dividing by the number of scores. In this case, there are six test scores. Thus,

Questions & Answers

find the domain and range of f(x)= 4x-7/x²-6x+8
Nick Reply
find the range of f(x)=(x+1)(x+4)
Jane Reply
-1, -4
Marcia
That's domain. The range is [-9/4,+infinity)
Jacob
If you're using calculus to find the range, you have to find the extrema through the first derivative test and then substitute the x-value for the extrema back into the original equation.
Jacob
Good morning,,, how are you
Harrieta Reply
d/dx{1/y - lny + X^3.Y^5}
mogomotsi Reply
How to identify domain and range
Umar Reply
hello
Akpevwe
He,,
Harrieta
hi
Dr
hello
velocity
I only talk to girls
Dr
women are smart then guys
Dr
Smarter
Adri
sorry
Dr
hi adri ana
Dr
:(
Shun
was up
Dr
hello
Adarsh
is it chatting app?.. I do not see any calculus here. lol
Adarsh
Find the arc length of the graph of f(x) = In (sinx) on the interval [Π/4, Π/2].
mukul Reply
Sand falling freely from a lorry form a conical shape whose height is always equal to one-third the radius of the base. a. How fast is the volume increasing when the radius of the base is (1m) and increasing at the rate of 1/4cm/sec Pls help me solve
ade
show that lim f(x) + lim g(x)=m+l
BARNABAS Reply
list the basic elementary differentials
Chio Reply
Differentiation and integration
Okikiola Reply
yes
Damien
proper definition of derivative
Syed Reply
the maximum rate of change of one variable with respect to another variable
Amdad
terms of an AP is 1/v and the vth term is 1/u show that the sum of uv terms is 1/2(uv+1)
Inembo Reply
what is calculus?
BISWAJIT Reply
calculus is math that studies the change in math, such as the rate and distance,
Tamarcus
what are the topics in calculus
Augustine
what is limit of a function?
Geoffrey Reply
what is x and how x=9.1 take?
Pravin Reply
what is f(x)
Inembo Reply
the function at x
Marc
also known as the y value so I could say y=2x or f(x)= 2x same thing just using functional notation your next question is what is dependent and independent variables. I am Dyslexic but know math and which is which confuses me. but one can vary the x value while y depends on which x you use. also
Marc
up domain and range
Marc
enjoy your work and good luck
Marc
I actually wanted to ask another questions on sets if u dont mind please?
Inembo
I have so many questions on set and I really love dis app I never believed u would reply
Inembo
Hmm go ahead and ask you got me curious too much conversation here
Adri
am sorry for disturbing I really want to know math that's why *I want to know the meaning of those symbols in sets* e.g n,U,A', etc pls I want to know it and how to solve its problems
Inembo
and how can i solve a question like dis *in a group of 40 students, 32 offer maths and 24 offer physics and 4 offer neither maths nor physics , how many offer both maths and physics*
Inembo
next questions what do dy mean by (A' n B^c)^c'
Inembo
The sets help you to define the function. The function is like a magic box where you put inside stuff(numbers or sets) and you get out the stuff but in different shapes (forms).
Adri
I dont understand what you wanna say by (A' n B^c)^c'
Adri
(A' n B (rise to the power of c)) all rise to the power of c
Inembo
Aaaahh
Adri
Ok so the set is formed by vectors and not numbers
Adri
A vector of length n
Adri
But you can make a set out of matrixes as well
Adri
I I don't even understand sets I wat to know d meaning of all d symbolsnon sets
Inembo
Wait what's your math level?
Adri
High-school?
Adri
yes
Inembo
am having big problem understanding sets more than other math topics
Inembo
So f:R->R means that the function takes real numbers and provides real numer. For ex. If f(x) =2x this means if you give to your function a real number like 2,it gives you also a real number 2times2=4
Adri
pls answer this question *in a group of 40 students, 32 offer maths and 24 offer physics and 4 offer neither maths nor physics , how many offer both maths and physics*
Inembo
If you have f:R^n->R^n you give to your function a vector of length n like (a1,a2,...an) where all a1,.. an are reals and gives you also a vector of length n... I don't know if i answering your question. Otherwise on YouTube you havr many videos where they explain it in a simple way
Adri
I would say 24
Adri
Offer both
Adri
Sorry 20
Adri
Actually you have 40 - 4 =36 who offer maths or physics or both.
Adri
I know its 20 but how to prove it
Inembo
You have 32+24=56who offer courses
Adri
56-36=20 who give both courses... I would say that
Adri
solution: In a question involving sets and Venn diagram, the sum of the members of set A + set B - the joint members of both set A and B + the members that are not in sets A or B = the total members of the set. In symbolic form n(A U B) = n(A) + n (B) - n (A and B) + n (A U B)'.
Mckenzie
In the case of sets A and B use the letters m and p to represent the sets and we have: n (M U P) = 40; n (M) = 24; n (P) = 32; n (M and P) = unknown; n (M U P)' = 4
Mckenzie
Now substitute the numerical values for the symbolic representation 40 = 24 + 32 - n(M and P) + 4 Now solve for the unknown using algebra: 40 = 24 + 32+ 4 - n(M and P) 40 = 60 - n(M and P) Add n(M and P), as well, subtract 40 from both sides of the equation to find the answer.
Mckenzie
40 - 40 + n(M and P) = 60 - 40 - n(M and P) + n(M and P) Solution: n(M and P) = 20
Mckenzie
thanks
Inembo
Simpler form: Add the sums of set M, set P and the complement of the union of sets M and P then subtract the number of students from the total.
Mckenzie
n(M and P) = (32 + 24 + 4) - 40 = 60 - 40 = 20
Mckenzie
Practice Key Terms 8

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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