MHT CET · Maths · Definite Integration
If \(\int_a^b x^3 d x=0\) and if \(\int_a^b x^2 d x=\frac{2}{3}\), then \(a\) and \(b\) are respectively
- A \(1,-1\)
- B \(-1,-1\)
- C 1,1
- D -1,1
Answer & Solution
Correct Answer
(D) -1,1
Step-by-step Solution
Detailed explanation
\(\begin{aligned} & \int_a^b x^3 d x=0 \text { and } \int_a^b x^2 d x=\frac{2}{3} \\ & \Rightarrow\left[\frac{x^4}{4}\right]_a^b=0 \text { and }\left[\frac{x^3}{3}\right]_a^b=\frac{2}{3} \\ & \Rightarrow \frac{b^4}{4}-\frac{a^4}{4}=0 \text { and } \frac{b^3}{3}-\frac{a^3}{3}=\frac{2}{3} \\ & \Rightarrow a^4=b^4 \text { and } b^3-a^3=2\end{aligned}\)
which is satisfied by \(\mathrm{a}=-1\) and \(\mathrm{b}=1\)
which is satisfied by \(\mathrm{a}=-1\) and \(\mathrm{b}=1\)
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