Surface integrals using a parametric description Evaluate the surface integral ∬ S f ( x , y , z ) d S using a parametric description of the surface . 28 . f ( x , y , z ) = y , where S is the cylinder x 2 + y 2 = 9 , 0 ≤ z ≤ 3
Surface integrals using a parametric description Evaluate the surface integral ∬ S f ( x , y , z ) d S using a parametric description of the surface . 28 . f ( x , y , z ) = y , where S is the cylinder x 2 + y 2 = 9 , 0 ≤ z ≤ 3
Surface integrals using a parametric descriptionEvaluate the surface integral
∬
S
f
(
x
,
y
,
z
)
d
S
using a parametric description of the surface.
28.
f
(
x
,
y
,
z
)
=
y
, where S is the cylinder
x
2
+
y
2
=
9
,
0
≤
z
≤
3
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
17. Suppose we know that the graph below is the graph of a solution to dy/dt = f(t).
(a) How much of the slope field can
you sketch from this information?
[Hint: Note that the differential
equation depends only on t.]
(b) What can you say about the solu-
tion with y(0) = 2? (For example,
can you sketch the graph of this so-
lution?)
y(0) = 1
y
AN
(b) Find the (instantaneous) rate of change of y at x = 5.
In the previous part, we found the average rate of change for several intervals of decreasing size starting at x = 5. The instantaneous rate of
change of fat x = 5 is the limit of the average rate of change over the interval [x, x + h] as h approaches 0. This is given by the derivative in the
following limit.
lim
h→0
-
f(x + h) − f(x)
h
The first step to find this limit is to compute f(x + h). Recall that this means replacing the input variable x with the expression x + h in the rule
defining f.
f(x + h) = (x + h)² - 5(x+ h)
=
2xh+h2_
x² + 2xh + h² 5✔
-
5
)x - 5h
Step 4
-
The second step for finding the derivative of fat x is to find the difference f(x + h) − f(x).
-
f(x + h) f(x) =
= (x²
x² + 2xh + h² -
])-
=
2x
+ h² - 5h
])x-5h) - (x² - 5x)
=
]) (2x + h - 5)
Macbook Pro
Evaluate the integral using integration by parts.
Sx² cos
(9x) dx
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