prove-that-0-1-1-x-7-1-3-dx-0-1-1-x-3-1-7-dx-

Question Number 76680 by Tony Lin last updated on 29/Dec/19

p r o v e t h a t :

\int_{0}^{1} {(1 - x^{7})}^{\frac{1}{3}} d x = \int_{0}^{1} {(1 - x^{3})}^{\frac{1}{7}} d x

Commented by mr W last updated on 29/Dec/19

You can't use 'macro parameter character #' in math mode

\int_{a}^{b} y (x) d x = \int_{c}^{d} x (y) d y

y = {(1 - x^{7})}^{\frac{1}{3}}

\Rightarrow y^{3} = 1 - x^{7}

\Rightarrow x = {(1 - y^{3})}^{\frac{1}{7}}

y (a = 0) = 1 = d

y (b = 1) = 0 = c

\int_{0}^{1} y (x) d x = \int_{0}^{1} {(1 - x^{7})}^{\frac{1}{3}} d x

= \int_{0}^{1} x (y) d y = \int_{0}^{1} {(1 - y^{3})}^{\frac{1}{7}} d y

= \int_{0}^{1} {(1 - x^{3})}^{\frac{1}{7}} d x

Commented by mr W last updated on 29/Dec/19

b o t h i n t e g r a l s r e p r e s e n t t h e s a m e

s h a d e d a r e a :

Commented by mr W last updated on 29/Dec/19

Commented by Tony Lin last updated on 29/Dec/19

t h a n k s s i r