# Calculate the area bounded by the polar curve #r=cos theta#?

# pi/4 #

The area we seek is.

If we convert to Polar Coordinates then the region

an angle from

#theta=-pi/2# to#theta=pi/2#

a ray from#r=0# to#r=cos theta#

And as we convert to Polar coordinates we get:

# x \ \ \ = rcos theta#

# y \ \ \ = rsin theta#

# dA = dy dy = r dr d theta#

So then the bounded area is given by#

# A = int int_R \ dA #

# \ \ = int_(-pi/2)^(pi/2) \ int_0^(cos theta) \ r dr d theta #

# \ \ = int_(-pi/2)^(pi/2) \ [1/2r^2]_0^(cos theta) \ d theta #

# \ \ = int_(-pi/2)^(pi/2) \ 1/2cos^2theta \ d theta #

# \ \ = int_(-pi/2)^(pi/2) \ 1/2 (1+cos2theta)/2 \ d theta #

# \ \ = 1/4 \ int_(-pi/2)^(pi/2) \ 1+cos2theta \ d theta #

# \ \ = 1/4 \ [theta + (sin2theta)/2]_(-pi/2)^(pi/2) #

# \ \ = 1/4 \ { (pi/2-1) - (-pi/2-(-1)) } #

# \ \ = 1/4 \ ( pi/2-1+pi/2+1) #

# \ \ = 1/4(pi) #

Observations:

Note that

# A = pi(1/2)^2 = pi/4#

Also, note that if we apply the double integration progress to a generic function

# A =int_alpha^beta \ 1/2r^2 \ theta # .

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To calculate the area bounded by the polar curve ( r = \cos(\theta) ), you need to evaluate the definite integral of ( \frac{1}{2} r^2 ) with respect to ( \theta ) over the interval where ( r = \cos(\theta) ) is positive. This interval is typically from ( \theta = 0 ) to the first positive root of ( r = \cos(\theta) ), which is ( \theta = \frac{\pi}{2} ).

So, the integral to compute the area is:

[ A = \int_{0}^{\frac{\pi}{2}} \frac{1}{2} (\cos(\theta))^2 , d\theta ]

After integrating, you'll get the value of the area ( A ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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