How do you find the area under the graph of #f(x)=cos(x)# on the interval #[-pi/2,pi/2]# ?

Answer 1
This is an integration problem. We will find the area under the curve #cos(x)# over the interval #[-pi/2,pi/2]#. This is inclusive because of the square brackets.
On the unit circle remember that the positive side of y-axis corresponds to #pi/2# and a coordinate of #(0,1).# The y-coordinate corresponds to #1.#
On the unit circle remember that the negative side of y-axis corresponds to #-pi/2# and a coordinate of #(0,-1).# The y-coordinate corresponds to #-1.#
#int_(-pi/2)^(pi/2)cos(x) dx#
#=[sin(x)]_(-pi/2)^(pi/2)=[sin(pi/2)-sin(-pi/2)]=1-(-1)=2#

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Answer 2

To find the area under the graph of ( f(x) = \cos(x) ) on the interval ([- \frac{\pi}{2}, \frac{\pi}{2}]), you need to integrate ( \cos(x) ) with respect to ( x ) from ( -\frac{\pi}{2} ) to ( \frac{\pi}{2} ). This can be done using the definite integral formula:

[ \text{Area} = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos(x) , dx ]

Integrating ( \cos(x) ) yields:

[ \int \cos(x) , dx = \sin(x) + C ]

Evaluating the definite integral:

[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos(x) , dx = \sin\left(\frac{\pi}{2}\right) - \sin\left(-\frac{\pi}{2}\right) = 1 - (-1) = 2 ]

So, the area under the graph of ( f(x) = \cos(x) ) on the interval ([- \frac{\pi}{2}, \frac{\pi}{2}]) is (2).

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Answer from HIX Tutor

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.

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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