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.
Determine whether the lines
L₁ (t) = (-2,3, −1)t + (0,2,-3) and
L2 p(s) = (2, −3, 1)s + (-10, 17, -8)
intersect. If they do, find the point of intersection.
Convert the line given by the parametric equations y(t)
Enter the symmetric equations in alphabetic order.
(x(t)
= -4+6t
= 3-t
(z(t)
=
5-7t
to symmetric equations.
Find the point at which the line (t) = (4, -5,-4)+t(-2, -1,5) intersects the xy plane.
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