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How do you find the area of the region bounded by the graph of #f(x)=x^4# and the #x#-axis on the interval #[-2,2]# ?

Answer 1

Charles Arocha

In this an integration problem. We need to find the area under the curve #f(x)=x^4# over the interval #-2# to #2#.

#int_-2^2x^4 dx = [x^5/5]_-2^2#

#=[(2)^5/5-(-2)^5/5]=[32/5-(-32)/5]#

#=[32/5+32/5]=[64/5]=12.8#

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Answer 2

Scarlett Allison

To find the area of the region bounded by the graph of ( f(x) = x^4 ) and the x-axis on the interval ([-2, 2]), you need to integrate the absolute value of the function ( f(x) ) over the interval ([-2, 2]). This is because the function ( f(x) = x^4 ) is always positive or zero on this interval.

The integral can be set up as follows:

[ \text{Area} = \int_{-2}^{2} |f(x)| , dx = \int_{-2}^{2} |x^4| , dx ]

[ = \int_{-2}^{2} x^4 , dx ]

Now, integrate ( x^4 ) over the interval ([-2, 2]) to find the area.

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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.