If \( \int_0^a\displaystyle\sqrt{x}\,dx=\displaystyle\frac{4a}{3},\ a\gt 0 \), then the value of \( \int_a^{a+1}\displaystyle x\,dx \) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( \displaystyle\frac{9}{2} \)
The given condition fixes a:
\( \displaystyle \int_0^a\sqrt{x}\,dx=\frac{2}{3}a^{3/2} \)
\( \displaystyle \frac{2}{3}a^{3/2}=\frac{4a}{3} \)
\( \displaystyle a^{1/2}=2 \)
\( a=4 \)
Then
\( \displaystyle \int_4^5x\,dx=\frac{25-16}{2} \)
\( \displaystyle =\frac{9}{2} \)