∫(7-|x|)dx
****Note it has limits from [-3,3]
I'm not looking the collection of antiderivatices with constant C, i'm looking for the Area beneath that function on top of x axis from[-3,3]
****Note it has limits from [-3,3]
I'm not looking the collection of antiderivatices with constant C, i'm looking for the Area beneath that function on top of x axis from[-3,3]
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So we can split the integral into ∫7 - ∫|x|
Then since |x| is an even function (that is, it is symmetric about the x-axis) and the limits of integration are symmetric, we can say ∫(-3,3) |x| dx = 2∫(0,3) |x| dx
But on the interval (0,3), we know that |x| = x.
So the whole thing becomes ∫(-3,3) 7dx - 2∫(0,3) xdx
But these are just power rules:
∫7 = 7x
∫x = .5x^2
Then plugging in the limits gives
= (7*3-7*(-3)) - 2(1/2)(3^2 - 0^2)
= (42) - (1)(9)
= 33.
Hope this is helpful.
Then since |x| is an even function (that is, it is symmetric about the x-axis) and the limits of integration are symmetric, we can say ∫(-3,3) |x| dx = 2∫(0,3) |x| dx
But on the interval (0,3), we know that |x| = x.
So the whole thing becomes ∫(-3,3) 7dx - 2∫(0,3) xdx
But these are just power rules:
∫7 = 7x
∫x = .5x^2
Then plugging in the limits gives
= (7*3-7*(-3)) - 2(1/2)(3^2 - 0^2)
= (42) - (1)(9)
= 33.
Hope this is helpful.