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

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::::::<math>y=1-x^2,~~~0\leq x \leq 1</math>
 
::::::<math>y=1-x^2,~~~0\leq x \leq 1</math>
  
::<span class="exam">is rotated about the <math style="vertical-align: -3px">y</math>-axis. Find the area of the resulting surface.
+
:::<span class="exam">is rotated about the <math style="vertical-align: -3px">y</math>-axis. Find the area of the resulting surface.
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"

Revision as of 12:25, 18 April 2016

a) Find the length of the curve
b) The curve
is rotated about the -axis. Find the area of the resulting surface.
Foundations:  
Recall:
1. The formula for the length of a curve where is
2.
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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