Improper integrals Improper integrals arise in polar coordinates when the radial coordinate r becomes arbitrarily large. Under certain conditions, these integrals are treated in the usual way: ∫ α β ∫ a ∞ f ( r , θ ) r d r d θ = lim b → ∞ ∫ α β ∫ a b f ( r , θ ) r d r d θ . Use this technique to evaluate the following integrals. 66. ∬ R d A ( 1 + x 2 + y 2 ) 2 ; R is the first quadrant.
Improper integrals Improper integrals arise in polar coordinates when the radial coordinate r becomes arbitrarily large. Under certain conditions, these integrals are treated in the usual way: ∫ α β ∫ a ∞ f ( r , θ ) r d r d θ = lim b → ∞ ∫ α β ∫ a b f ( r , θ ) r d r d θ . Use this technique to evaluate the following integrals. 66. ∬ R d A ( 1 + x 2 + y 2 ) 2 ; R is the first quadrant.
Solution Summary: The author evaluates the value of the given integral. The region is located in the first quadrant.
Improper integralsImproper integrals arise in polar coordinates when the radial coordinate r becomes arbitrarily large. Under certain conditions, these integrals are treated in the usual way:
∫
α
β
∫
a
∞
f
(
r
,
θ
)
r
d
r
d
θ
=
lim
b
→
∞
∫
α
β
∫
a
b
f
(
r
,
θ
)
r
d
r
d
θ
.
Use this technique to evaluate the following integrals.
66.
∬
R
d
A
(
1
+
x
2
+
y
2
)
2
;
R is the first quadrant.
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.
(2) Evaluate the iterated integral by converting to polar coordinates.
1 Vx-x2
|| (x² + y²) dx dy
(1 point) Evaluate the iterated integral by converting to polar coordinates.
NOTE: When typing your answers use "th" for 0.
/6-y2
2x + 4y dx dy
Σ
dr de
=
where
a =
Σ
b =
pi/2
Σ
c =
Σ
d =
6-y2
2x + 4y dx dy =
Σ
M M MM
(3) Evaluate the iterated integral by converting to polar coordinates.
4-y2
(x + y) dx dy
0,
Chapter 16 Solutions
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