Question

Please help me solve this Question 1)Y=x^6/5 , y=4x^1/5

Find the area between the curves


…The answer they want looks similar to this (ex. 123/455 • 12^11/10)

Thank you so much I really appreciate it

Answer 1

`(200)/(33) \cdot 4^{(1)/(5) }`

Explanation:

We have :

`y = x^{(6)/(5) } \,\,\, and \,\,\, y = 4x^{(1)/(5) }`

`y = x^{(6)/(5) } = 4x^{(1)/(5) }\\\\\implies x^{(6)/(5) } = 4x^{(1)/(5) }\\\\\implies 4x^{(1)/(5) } - x^{(6)/(5) } = 0\\\\\implies x^{(1)/(5) }[4 - x] = 0\\\\\implies x^{(1)/(5) } = 0 \;\;\;or\;\;\;4-x=0\\\\\implies x = 0 \;\;\;or\;\;\; x = 4`

Area between the curves:

`\int\limits^4_0 {4x^{(1)/(5) }} \, dx -\int\limits^4_0 {x^{(6)/(5) }} \, dx \\\\=4\int\limits^4_0 {x^{(1)/(5) }} \, dx -\int\limits^4_0 {x^{(6)/(5) }} \, dx \\\\=4[\frac{x^{^{(1)/(5) +1}}}{(1)/(5) +1} ]_(_0)^(^4)-[\frac{x^{^{(6)/(5) +1}}}{(6)/(5) +1} ]_(_0)^(^4)\\\\=4[\frac{x^{^{(6)/(5)}}}{(6)/(5)} ]_(_0)^(^4) - [\frac{x^{^{(11)/(5)}}}{(11)/(5)} ]_(_0)^(^4)\\\\`

`= 4((5)/(6))[x^{^{(6)/(5)}}]_(_0)^(^4) - (5)/(11) [x^{^{(11)/(5)}}]_(_0)^(^4)\\\\=(10)/(3)[4^{^{(6)/(5)}}-0^{^{(6)/(5)}}] -(5)/(11)[4^{^{(11)/(5)}}-0^{^{(11)/(5)}}] \\\\=(10)/(3) [4^{^{(5)/(5)}} * 4^{^{(1)/(5)}}] - (5)/(11) [4^{^{(10)/(5)}} * 4^{^{(1)/(5)}}]\\\\=(10)/(3) [4 * 4^{^{(1)/(5)}}] - (5)/(11) [4^{^(2)} * 4^{^{(1)/(5)}}]\\\\`

`= 4^{^{(1)/(5)}}[(10*4)/(3) - (5*4^2)/(11) ]\\\\= 4^{^{(1)/(5)}}[(40)/(3) - (80)/(11) ]\\\\= 4^{^{(1)/(5)}}[(40(11) - 80(3))/(3(11)) ]\\\\= 4^{^{(1)/(5)}}[(440 - 240)/(33) ]\\\\= 4^{^{(1)/(5)}}[(200)/(33) ]`