Interesting Integrals 4

Calculus Level 3

P = ∫ 0 1 2 x 2 d x , Q = ∫ 0 1 2 x 3 d x , R = ∫ 1 2 2 x 2 d x , S = ∫ 1 2 2 x 3 d x \large P= \int_0^1 2^{x^2} \, dx ,\quad Q= \int_0^1 2^{x^3} \, dx ,\quad R= \int_1^2 2^{x^2} \, dx ,\quad S= \int_1^2 2^{x^3} \, dx

Given the four equations above. Which of the following inequalities is true?

Q > P Q>P R = S R=S P > Q P>Q R > S R>S

Krishna Shankar by Krishna Shankar , 23, India

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2 solutions

We note that both x 2 x^2 and x 3 x^3 increase with x x and for { 0 ≤ x ≤ 1 : x 2 ≥ x 3 ⟹ 2 x 2 ≥ 2 x 3 x > 1 : x 2 < x 3 ⟹ 2 x 2 < 2 x 3 ⟹ Q < P < R < S \begin{cases} 0 \le x \le 1: & x^2 \ge x^3 & \implies 2^{x^2} \ge 2^{x^3} \\ x > 1: & x^2 < x^3 & \implies 2^{x^2} < 2^{x^3} \end{cases} \implies \color{#3D99F6}{Q < P} < R < S

Therefore, the answer is P > Q \boxed{P>Q} .

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Sabhrant Sachan
Jul 9, 2016

consider f ( x ) = 2 x 3 and g ( x ) = 2 x 2 f ( x ) > g ( x ) for x ∈ ( 1 , 2 ) ⟹ S > R f ( x ) < g ( x ) for x ∈ ( 0 , 1 ) ⟹ P > Q \text{ consider } f(x) = 2^{x^3} \text{ and } g(x)=2^{x^2} \\ f(x) > g(x) \text{ for } x \in (1,2) \implies S > R \\ f(x) < g(x) \text{ for } x \in (0,1) \implies\boxed{ P>Q}

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