Area Consider the three regions A , B . and C determined by the graph of f ( x ) = arcsin x , as shown in the figure. (a) Calculate the areas of regions A and B. (b) Use your answers in part (a) to evaluate the integral ∫ 1 / 2 2 / 2 arcsin x d x . (c) Use tlie methods in part (a) to evaluate the integral ∫ 1 3 ln x d x . (d) Use the methods in part (a) to evaluate the integral ∫ 1 3 arctan x d x .
Area Consider the three regions A , B . and C determined by the graph of f ( x ) = arcsin x , as shown in the figure. (a) Calculate the areas of regions A and B. (b) Use your answers in part (a) to evaluate the integral ∫ 1 / 2 2 / 2 arcsin x d x . (c) Use tlie methods in part (a) to evaluate the integral ∫ 1 3 ln x d x . (d) Use the methods in part (a) to evaluate the integral ∫ 1 3 arctan x d x .
Solution Summary: The author calculates the area of the regions A,B,C determined by the graph of f(x)=mathrmsin-1x.
Area Consider the three regions A, B. and C determined by the graph of
f
(
x
)
=
arcsin
x
, as shown in the figure.
(a) Calculate the areas of regions A and B.
(b) Use your answers in part (a) to evaluate the integral
∫
1
/
2
2
/
2
arcsin
x
d
x
.
(c) Use tlie methods in part (a) to evaluate the integral
∫
1
3
ln
x
d
x
.
(d) Use the methods in part (a) to evaluate the integral
∫
1
3
arctan
x
d
x
.
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.
Let V be the volume of the solid obtained by rotating about the y-axis the region bounded y = √16x and y
V =
Draw a diagram to explain your method.
15
10
5
y
15
10
5
y
=
Find V by slicing.
16
X
О
-15 -10
-5
5
10
15
О
-15
-10
-5
5
10
15
15
10
y
15
10
5
y
x
-15
-10
-5
5
10
-15 -10
-5
5
10
15
10
X
15
a) let SSK : A->R be function and let
c be acluster Point of A if lim S, (x) exists
for each i=1, 2, .-,k then
K
i) lim Si (x)= lim fi (x)
X->C 1=1
11), im π fi (x) = lim fi (x)
YC il
i=1
1) let f(x) = ) x² Sin (1/x), xe Q/{o}
f(x) = {
x² cos(\/x), x&Q
Show that lim f(x)= 0
X = 0
c) Give an example of aset ASR, a cluster Point C
of Aand two fun. & 9: AR st lim f(x)9(x) exsis
bat limfex) does not exist
X-C
2. [-/4 Points]
DETAILS
MY NOTES
SESSCALCET2 7.3.002.
Let S be the solid obtained by rotating the region shown in the figure about the y-axis. (Assume a = 6 and b = 2.)
ASK YOUR TEACHER
0
y = a sin(bx²)
Sketch a typical approximating shell.
y
6
4
2
x
π/b
y
2
1
x
0.5
1.0
1.5
0.2
0.4
0.6
0.8
1.0
-2
-1
-4
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