Definite integral and comparison theorem

Question

Use the comparison theorem to show that 

 

₀ ∫¹ x^5 dx ≤ ₀ ∫¹ x^4 dx , 

 

₁∫² x^4 dx ≤ ₁∫² x^5 dx 

Answer 1

Regarding your first inequality: on [0,1) the values of x⁵ are always less than x⁴ because raising a positive number to a higher power produces a lesser quantity; thinking of Riemann sums, each rectangle's height for the integration of the fifth power of x is less than its corresponding height when integrating the fourth power of x, and therefore the inequality holds (smaller area under curve x⁴.

 

Similar reasoning applies to your second inequality, based on the fact that x⁵ > x⁴ when x>1.