Question

i have some problem in this question​

Answer 1

Question:

If

$\displaystyle{\sf\:\int\limits_a^b\:x^3\:dx\:=\:0}$ and

$\displaystyle{\sf\:\int\limits_a^b\:x^2\:dx\:=\:\dfrac{2}{3}}$ then find the values of a and b.

Answer:

$\displaystyle{\boxed{\red{\sf\:(\:a\:,\:b\:)\:=\:(\:-\:1\:,\:1\:)\:}}}$

Step-by-step-explanation:

We have given two integrals.

We have to find the values of their lower limit, a and upper limit, b.

Now,

$\displaystyle{\sf\:\int\limits_a^b\:x^3\:dx\:=\:0}$

We know that,

$\displaystyle{\boxed{\pink{\sf\:\int\:x^n\:dx\:=\:\dfrac{x^{n\:+\:1}}{n\:+\:1}}}}$

$\displaystyle{\implies\sf\:\dfrac{x^{3\:+\:1}}{3\:+\:1}\:\bigg|_a^b\:=\:0}$

$\displaystyle{\implies\sf\:\dfrac{x^4}{4}\:\bigg|_a^b\:=\:0}$

Substituting upper and lower limits, we get,

$\displaystyle{\implies\sf\:\dfrac{b^4}{4}\:-\:\dfrac{a^4}{4}\:=\:0}$

$\displaystyle{\implies\sf\:\dfrac{b^4\:-\:a^4}{4}\:=\:0}$

$\displaystyle{\implies\sf\:b^4\:-\:a^4\:=\:0}$

$\displaystyle{\implies\sf\:b^4\:=\:a^4}$

$\displaystyle{\implies\sf\:\sqrt[4]{\sf\:b^4}\:=\:\sqrt[4]{\sf\:a^4}}$

$\displaystyle{\implies\boxed{\purple{\sf\:b\:=\:\pm\:a}}}$

Now,

$\displaystyle{\sf\:\int\limits_a^b\:x^2\:dx\:=\:\dfrac{2}{3}}$

We know that,

$\displaystyle{\boxed{\pink{\sf\:\int\:x^n\:dx\:=\:\dfrac{x^{n\:+\:1}}{n\:+\:1}}}}$

$\displaystyle{\implies\sf\:\dfrac{x^{2\:+\:1}}{2\:+\:1}\:\bigg|_a^b\:=\:\dfrac{2}{3}}$

$\displaystyle{\implies\sf\:\dfrac{x^3}{3}\:\bigg|_a^b\:=\:\dfrac{2}{3}}$

Substituting upper and lower limits, we get,

$\displaystyle{\implies\sf\:\dfrac{b^3}{3}\:-\:\dfrac{a^3}{3}\:=\:\dfrac{2}{3}}$

$\displaystyle{\implies\sf\:\dfrac{b^3\:-\:a^3}{\cancel{3}}\:=\:\dfrac{2}{\cancel{3}}}$

$\displaystyle{\implies\sf\:b^3\:-\:a^3\:=\:2}$

By using b = a, we get,

$\displaystyle{\implies\sf\:a^3\:-\:a^3\:=\:2}$

$\displaystyle{\implies\sf\:0\:\neq\:2}$

∴ b = a is not acceptable.

Now, using b = - a, we get,

$\displaystyle{\implies\sf\:(\:-\:a\:)^3\:-\:a^3\:=\:2}$

$\displaystyle{\implies\sf\:-\:a^3\:-\:a^3\:=\:2}$

$\displaystyle{\implies\sf\:-\:2\:a^3\:=\:2}$

$\displaystyle{\implies\sf\:a^3\:=\:-\:\cancel{\dfrac{2}{2}}}$

$\displaystyle{\implies\sf\:a^3\:=\:-\:1}$

$\displaystyle{\implies\:\boxed{\blue{\sf\:a\:=\:-\:1}}}$

Now,

$\displaystyle{\sf\:b\:=\:-\:a}$

$\displaystyle{\implies\sf\:b\:=\:-\:(\:-\:1\:)}$

$\displaystyle{\implies\:\boxed{\green{\sf\:b\:=\:1}}}$

$\displaystyle{\therefore\:\underline{\boxed{\red{\sf\:(\:a\:,\:b\:)\:=\:(\:-\:1\:,\:1\:)\:}}}}$