The value of a for which \(\int_{1}^{a}\displaystyle x^2\,dx=\frac{7}{3}\) is :
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 2
By the fundamental theorem of calculus:
\( \displaystyle \int_{1}^{a}x^2\,dx=\left[\frac{x^3}{3}\right]_{1}^{a}=\frac{a^3-1}{3} \)
Setting it equal to 7/3:
\(a^3-1=7\)
\(a^3=8\)
\(a=2\)
(a = 2 is the only real cube root of 8.)