By Stokes' theorem,

where

is the circular boundary of the hemisphere

in the

-

plane. We can parameterize the boundary via the "standard" choice of polar coordinates, setting

where

. Then the line integral is


We can check this result by evaluating the equivalent surface integral. We have

and we can parameterize

by

so that

where

and

. Then,

as expected.
Well first you do 6÷3=2. Then you do 2*5= 10. Also do 20+10=30-15=15. So your answer is 15
Answer: 166 2/3m
Step-by-step explanation:
Area of a rectangle = length times width
Length: 16 2/3
To find the width, multiply it by 3/5.
w = 16 2/3 · 3/5
w = 10
Then, multiply the length by the width.
a = l · w
a = 16 2/3 · 10
a = 166 2/3m
Answer:
See attachment for graph
Step-by-step explanation:
Given

Required
Plot the graph
First, we need to make a table.
To do this, we assume values for x
When 


When 


When 


When 


So, the table is:


<em>See attachment for graph</em>
Answer:
The Riemann Sum for
with n = 4 using midpoints is about 24.328125.
Step-by-step explanation:
We want to find the Riemann Sum for
with n = 4 using midpoints.
The Midpoint Sum uses the midpoints of a sub-interval:

where 
We know that a = 4, b = 5, n = 4.
Therefore, 
Divide the interval [4, 5] into n = 4 sub-intervals of length 
![\left[4, \frac{17}{4}\right], \left[\frac{17}{4}, \frac{9}{2}\right], \left[\frac{9}{2}, \frac{19}{4}\right], \left[\frac{19}{4}, 5\right]](https://tex.z-dn.net/?f=%5Cleft%5B4%2C%20%5Cfrac%7B17%7D%7B4%7D%5Cright%5D%2C%20%5Cleft%5B%5Cfrac%7B17%7D%7B4%7D%2C%20%5Cfrac%7B9%7D%7B2%7D%5Cright%5D%2C%20%5Cleft%5B%5Cfrac%7B9%7D%7B2%7D%2C%20%5Cfrac%7B19%7D%7B4%7D%5Cright%5D%2C%20%5Cleft%5B%5Cfrac%7B19%7D%7B4%7D%2C%205%5Cright%5D)
Now, we just evaluate the function at the midpoints:




Finally, use the Midpoint Sum formula

This is the sketch of the function and the approximating rectangles.