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