In the following exercises, evaluate the double integral ∬ D f ( x , y ) d A over the region D. 75. f ( x , y ) = 1 and D = { ( x , y ) | 0 ≤ x ≤ π 2 , sin x ≤ y ≤ 1 + sin x }
In the following exercises, evaluate the double integral ∬ D f ( x , y ) d A over the region D. 75. f ( x , y ) = 1 and D = { ( x , y ) | 0 ≤ x ≤ π 2 , sin x ≤ y ≤ 1 + sin x }
In the following exercises, evaluate the double integral
∬
D
f
(
x
,
y
)
d
A
over the region D. 75.
f
(
x
,
y
)
=
1
and
D
=
{
(
x
,
y
)
|
0
≤
x
≤
π
2
,
sin
x
≤
y
≤
1
+
sin
x
}
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.
(4) (10 points) Evaluate
√(x² + y² + z²)¹⁄² exp[}(x² + y² + z²)²] dV
where D is the region defined by 1< x² + y²+ z² ≤4 and √√3(x² + y²) ≤ z.
Note: exp(x² + y²+ 2²)²] means el (x²+ y²+=²)²]¸
(2) (12 points) Let f(x,y) = x²e¯.
(a) (4 points) Calculate Vf.
(b) (4 points) Given x
directional derivative
0, find the line of vectors u =
D₁f(x, y) = 0.
(u1, 2) such that the
-
(c) (4 points) Let u= (1+3√3). Show that
Duƒ(1, 0) = ¦|▼ƒ(1,0)| .
What is the angle between Vf(1,0) and the vector u? Explain.
Find the missing values by solving the parallelogram shown in the figure. (The lengths of the diagonals are given by c and d. Round your answers to two decimal places.)
a
b
29
39
66.50
C
17.40
d
0
54.0
126°
a
Ꮎ
b
d
University Calculus: Early Transcendentals (4th Edition)
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