Verify the formulas in Problem 21 to 27 by contour integration or as indicated. Assume a > 0 , m > 0. ∫ 0 2 π d θ ( a + b sin θ ) 2 = ∫ 0 2 π d θ ( a + b cos θ ) 2 = 2 π a a 2 − b 2 3 / 2 , | b | < a Hint: You can do this directly by contour integration, but it is easier to differentiate Problem 21 with respect to a .
Verify the formulas in Problem 21 to 27 by contour integration or as indicated. Assume a > 0 , m > 0. ∫ 0 2 π d θ ( a + b sin θ ) 2 = ∫ 0 2 π d θ ( a + b cos θ ) 2 = 2 π a a 2 − b 2 3 / 2 , | b | < a Hint: You can do this directly by contour integration, but it is easier to differentiate Problem 21 with respect to a .
Verify the formulas in Problem 21 to 27 by contour integration or as indicated. Assume
a
>
0
,
m
>
0.
∫
0
2
π
d
θ
(
a
+
b
sin
θ
)
2
=
∫
0
2
π
d
θ
(
a
+
b
cos
θ
)
2
=
2
π
a
a
2
−
b
2
3
/
2
,
|
b
|
<
a
Hint: You can do this directly by contour integration, but it is easier to differentiate Problem 21 with respect to
a
.
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.
1. Find the points at which r = 2 cos 2θ and r=1 intersect.
4. Find / 4 (36x? +20x) 7 6r+Sx° dx.
Solve the following:
a. If z = sin(5x + 3y) Find and
dx
dz
ду
b. If z=(3x+2y) (4x-5y) Find Oz
dz
and
8x by
c. A cylinder has dimensions r=5 cm, h=12 cm. Find the approximate increase in
volume when r increases by 0.2 cm and h decreases by 0.3 cm.
d. The radius of a cylinder increases at the rate of 0.2 cm/s while the height
decreases at the rate of 0.4 cm/s. Find the rate at which the volume is changing
at the instant when r = 8 cm and h = 12 cm.
e. If z=sin(x + y); x = u² + v²; y = 2uv Find
dz
and
อน
av
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