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Question

If $$\displaystyle \int _{ a }^{ b }{ { x }^{ 3 } } dx=0$$ and $$\displaystyle \int _{ a }^{ b }{ { x }^{ 2 }dx } =\cfrac { 2 }{ 3 } $$, then what are the values of $$a$$ and $$b$$ respectively?


A
1,1
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B
1,1
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C
0,0
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D
2,2
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Solution

The correct option is A $$-1,1$$
from given equation , we can get

       $$b^4-a^4=0\Rightarrow (b-a)(b+a)(b^2+a^2)=0$$...............(1)

and $$\dfrac{b^3-a^3}3=\dfrac23\Rightarrow b^3-a^3=2\Rightarrow (b-a)(b^2+ab+a^2)=2$$...............(2)

From (2) we can see that $$(b-a)$$ is not equal to 0 and $$(b^2+a^2)$$ cannot be zero as it is always positive in (1)

so, in (1) we can see that $$(b+a)=0\Rightarrow a=-b$$

Putting $$a=-b$$ in (2)
$$2b\cdot b^2=2\Rightarrow b=1\ or\ -1$$

so. $$a=-1\ or \ 1$$

Therefore, Answer is $$A$$

Mathematics

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