Calculas homework
aaa6j
222222.pdf
#1 below is to be turned in as a graded hw quiz AT THE FIRST OF CLASS ON TUESDAY.
#2 below is to be turned in as a second graded hw quiz AT THE FIRST OF CLASS ON TUESDAY.
Note that the first thing you need to do is draw a graph showing the function over the interval.
YOU MUST SHOW ALL STEPS TO RECEIVE CREDIT.
1. The area between the graph of 2
)( xxf and the x-axis on the interval [0,4] is the definite
integral: dxx 4
0
2 . Using the shortcut (fundamental theorem of calculus) we know
3
64
3
0
3
4
3
33 4
0
34
0
2
x dxx . Your assignment is to use the definition of definite integral
xxfdxxf n
i
i n
b
a
)(lim)(
1
to evaluate dxx 4
0
2 in order to find the area between the graph of
2 )( xxf and the x-axis on the interval [0,4]. Recall that this is a five step process: Partition
[a,b] into n subintervals. Focus on the ith subinterval. Pick a number i
x in the ith subinterval
(picking the right hand endpoint i
x is the easiest. Write an expression for the area of the ith
rectangle. Write an expression for the sum of the areas of the n rectangles using summation
notation. Take the limit as n approaches infinity. Details provided in class. The result will be
3
64
2. The area between the graph of 3
)( xxf and the x-axis on the interval [0,1] is the definite
integral: dxx 1
0
3 . Using the shortcut (fundamental theorem of calculus) we find this has a value
of ¼. Your assignment is to use the definition of definite integral
xxfdxxf n
i
i n
b
a
)(lim)(
1
to evaluate dxx 1
0
3 in order to find the area between the graph of
3 )( xxf and the x-axis on the interval [0,1]. The result will be
4
1
