Writing a Limit as a Definite Integral In Exercises 11-14, write the limit as a definite integral on the given interval, where c i is any point in the ith subinterval. lim ‖ Δ ‖ → 0 ∑ i = 1 n ( 2 − c i sin c i ) Δ x i , [ 0 , π ]
Writing a Limit as a Definite Integral In Exercises 11-14, write the limit as a definite integral on the given interval, where c i is any point in the ith subinterval. lim ‖ Δ ‖ → 0 ∑ i = 1 n ( 2 − c i sin c i ) Δ x i , [ 0 , π ]
Solution Summary: The author explains the formula for calculating the definite integral of f(x).
Writing a Limit as a Definite Integral In Exercises 11-14, write the limit as a definite integral on the given interval, where
c
i
is any point in the ith subinterval.
lim
‖
Δ
‖
→
0
∑
i
=
1
n
(
2
−
c
i
sin
c
i
)
Δ
x
i
,
[
0
,
π
]
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
Part (b)
Define a functiong : [0, 1) R by
tan(tx)
g (1) :=
-dx
x3/2
for all i € [0, 1).
(i)
Show that g is continuous on [0, 1).
(ii)
Show that g is differentiable on [0, 1).
Left- and right-continuity
a. Is hlx2 = Vx - 9 left-continuous at x = 3? Explain.
b. Is h1x2 = Vx - 9 right-continuous at x = 3? Explain.
Assignment Real Analysis (2): handwriting
1. Give an example of a bounded function which is not Riemann integrable over
[0,1).
2. Let f(x) on [0,1]. Show that fe R(0, 1] and find f f.
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