Difference between revisions of "009B Sample Final 1, Problem 7"

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!Step 3:  
 
!Step 3:  
 
|-
 
|-
|Now, we use <math>u</math>-substitution. Let <math>u=\sec \theta</math>. Then, <math>du=\sec \theta \tan \theta d\theta</math>.  
+
|Now, we use <math>u</math>-substitution. Let <math style="vertical-align: 0px">u=\sec \theta</math>. Then, <math style="vertical-align: -1px">du=\sec \theta \tan \theta d\theta</math>.  
 
|-
 
|-
 
|So, the integral becomes
 
|So, the integral becomes

Revision as of 11:09, 22 February 2016

a) Find the length of the curve

.

b) The curve

is rotated about the -axis. Find the area of the resulting surface.

Foundations:  
1. The formula for the length of a curve where is
.
2. Recall that .
3. The surface area of a function rotated about the -axis is given by
where .

Solution:

(a)

Step 1:  
First, we calculate .
Since .
Using the formula given in the Foundations section, we have
.
Step 2:  
Now, we have:
Step 3:  
Finally,

(b)

Step 1:  
We start by calculating .
Since .
Using the formula given in the Foundations section, we have
.
Step 2:  
Now, we have
We proceed by using trig substitution. Let . Then, .
So, we have
Step 3:  
Now, we use -substitution. Let . Then, .
So, the integral becomes
Step 4:  
We started with a definite integral. So, using Step 2 and 3, we have
Final Answer:  
(a)
(b)

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