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.
Closed-curve integrals Evaluate ∮C ds, ∮C dx, and ∮C dy, where Cis the unit circle oriented counterclockwise.
Polar curves C2: r = 4cosθ and C3 : r = 4sinθ.
Solve for the slope of the tangent line at the non-pole point on C2 which intersects C3 .
Set up an integral giving the area of the region inside both C2 and C3 .
An ice cream cone can be modeled by the region bounded by thehemisphere z = √8 − x2 − y2 and the cone z = √x2 + y2. We wish to find the volumeof the ice cream cone.(a) sketch this ice cream cone in 3 dimensions. (b) Explain why a polar coordinates interpretation of this problem is advan-tageous. include a comparison to the double integral which results fromthe rectangular coordinates interpretation of this problem. (c) Restate using polar coordinates, r and θ, statethe bounds on each.(d) Set up and evaluate the double integral corresponding to the volume ofthe ice cream cone.
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