Definite Integration Question 487
Question: If $ \int_a^{b}{x^{3}dx}=0 $ and $ \int_a^{b}{x^{2}}dx=\frac{2}{3} $ , then the value of a and b will be respectively
[AMU 2005]
Options:
A) 1, 1
B) $ -1,-1 $
C) $ 1,-1 $
D) $ -1,1 $
Show Answer
Answer:
Correct Answer: D
Solution:
$ \int_a^{b}{x^{3}dx=0} $ , (Given)
$ . | \frac{1}{4}x^{4} . |_a^{b}=0 $
therefore $ \frac{1}{4}(b^{4}-a^{4})=0 $
therefore $ b^{4}-a^{4}=0 $
$ \int_a^{b}{x^{2}dx=\frac{2}{3}} $
therefore $ . | \frac{x^{3}}{3} . |_a^{b}=\frac{2}{3} $
$ \Rightarrow $ $ b^{3}-a^{3}=2 $
$ \Rightarrow $ $ b^{4}-a^{4}=0 $
therefore $ (b^{2}-a^{2})(b^{2}+a^{2})=0 $
or $ (b-a)(b+a)=0 $
$ \Rightarrow $ $ b=\pm a $
but $ b=a $ does not satisfy the equation,
$ \therefore b=-a $
Now, $ b^{3}-a^{3}=2 $
therefore $ {{(-a)}^{3}}-a^{3}=2 $
$ -2a^{3}=2 $ or $ a^{3}=-1 $
therefore $ a=-1 $
therefore $ b=-a $
Hence $ b=-(-1)=1 $
therefore $ a=-1 $ , $ b=1 $
therefore $ (a,b)=(-1,\ 1) $ .
BETA
