Use the Wallis sine and cosine formulas: ∫ 0 π / 2 sin n x d x = π 2 ⋅ 1 ⋅ 3 ⋅ 5 ⋅ ⋅ ⋅ n − 1 2 ⋅ 4 ⋅ 6 ⋅ ⋅ ⋅ n n even and ≥ 2 ∫ 0 π / 2 sin n x d x = 2 ⋅ 4 ⋅ 6 ⋅ ⋅ ⋅ n − 1 3 ⋅ 5 ⋅ 7 ⋅ ⋅ ⋅ n n odd and ≥ 3 ∫ 0 π / 2 cos n x d x = π 2 ⋅ 1 ⋅ 3 ⋅ 5 ⋅ ⋅ ⋅ n − 1 2 ⋅ 4 ⋅ 6 ⋅ ⋅ ⋅ n n even and ≥ 2 ∫ 0 π / 2 cos n x d x = 2 ⋅ 4 ⋅ 6 ⋅ ⋅ ⋅ n − 1 3 ⋅ 5 ⋅ 7 ⋅ ⋅ ⋅ n n odd and ≥ 3 Find the centroid of the solid bounded above by the paraboloid z = x 2 + y 2 , below by the plane z = 0 , and laterally by the cylinder x − 1 2 + y 2 = 1.
Use the Wallis sine and cosine formulas: ∫ 0 π / 2 sin n x d x = π 2 ⋅ 1 ⋅ 3 ⋅ 5 ⋅ ⋅ ⋅ n − 1 2 ⋅ 4 ⋅ 6 ⋅ ⋅ ⋅ n n even and ≥ 2 ∫ 0 π / 2 sin n x d x = 2 ⋅ 4 ⋅ 6 ⋅ ⋅ ⋅ n − 1 3 ⋅ 5 ⋅ 7 ⋅ ⋅ ⋅ n n odd and ≥ 3 ∫ 0 π / 2 cos n x d x = π 2 ⋅ 1 ⋅ 3 ⋅ 5 ⋅ ⋅ ⋅ n − 1 2 ⋅ 4 ⋅ 6 ⋅ ⋅ ⋅ n n even and ≥ 2 ∫ 0 π / 2 cos n x d x = 2 ⋅ 4 ⋅ 6 ⋅ ⋅ ⋅ n − 1 3 ⋅ 5 ⋅ 7 ⋅ ⋅ ⋅ n n odd and ≥ 3 Find the centroid of the solid bounded above by the paraboloid z = x 2 + y 2 , below by the plane z = 0 , and laterally by the cylinder x − 1 2 + y 2 = 1.
∫
0
π
/
2
sin
n
x
d
x
=
π
2
⋅
1
⋅
3
⋅
5
⋅
⋅
⋅
n
−
1
2
⋅
4
⋅
6
⋅
⋅
⋅
n
n
even
and
≥
2
∫
0
π
/
2
sin
n
x
d
x
=
2
⋅
4
⋅
6
⋅
⋅
⋅
n
−
1
3
⋅
5
⋅
7
⋅
⋅
⋅
n
n
odd
and
≥
3
∫
0
π
/
2
cos
n
x
d
x
=
π
2
⋅
1
⋅
3
⋅
5
⋅
⋅
⋅
n
−
1
2
⋅
4
⋅
6
⋅
⋅
⋅
n
n
even
and
≥
2
∫
0
π
/
2
cos
n
x
d
x
=
2
⋅
4
⋅
6
⋅
⋅
⋅
n
−
1
3
⋅
5
⋅
7
⋅
⋅
⋅
n
n
odd
and
≥
3
Find the centroid of the solid bounded above by the paraboloid
z
=
x
2
+
y
2
,
below by the plane
z
=
0
,
and laterally by the cylinder
x
−
1
2
+
y
2
=
1.
For the given graph, determine the following.
-3
12
УА
4
3
-
-1
°
1 2
3
x
-1.
-2-
a. Determine for which values of a the lim f (x) exists but f is not continuous at x = a.
a
b. Determine for which values of a the function is continuous but not differentiable at x = a.
a
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