can somebody show I how to get to the answer 256π/15.
This is how I solve for it basing on what the last tutor show I but got the wrong answer, so I don't know what I did wrong. The question is 2π∫ from 0 to 8 [y(√(y/2)-(y/4)]dy.
These are how I do but got the wrong answer
2π∫ from 0 to 8 [y(y^(1/2)/√2)-(y/4))]dy
2π∫ from 0 to 8 [(y^(3/2)/√2)-(y^2)/4]dy
2π [(2y^(5/2))/5√2)-(y^3)/12]from 0 to 8
2π [((2*8^(5/2))/5√2)-(8^3)/12)-((2*0^(5/2))/5√2)-(0^3)/12]
2π [((2*81.02)/5√2)-512/12)-(0-0)]
2π [(162.04/5√2)-(512/12)]
2π [(12*162.04)/(12*(5√2))-((5√2)*512)/((5√2)*12)]
2π [(1944.48/60√2-(2560√2)/60√2]
2π [-615.52/60√2]
1231.04π/60√2
Oh come on.
2π [(2*8^(5/2)/(5√2)-(8^3)/12)-((2*0^(5/2))/5√2)-(0^3)/12]
2π [256/5 - 512/12]
2π [3072/60 - 2560/60]
2π [512/60]
2π [128/15]
256π/15
(2*8^(5/2))/(5√2)
(2*2^(15/2))/(5√2)
(2*2^7*√2)/(5√2)
(2*128√2)/(5√2)
256/5
(2*8^(5/2))/(5√2) = 256/5
fix that and the rest follows
You're carrying a lot of extra √2 stuff around
can you show the steps from (2*8^(5/2))/(5√2) to = 256/5 steve?
To solve the given integral, let's go through each step together:
Step 1: Write out the integral expression:
∫(0 to 8) [y(√(y/2)-(y/4))]dy
Step 2: Simplify the expression inside the integral using the distributive property:
∫(0 to 8) [y * √(y/2)] - [y * (y/4)] dy
Step 3: Simplify the first term inside the integral:
∫(0 to 8) [(y * √y) / √2] - [(y^2)/4] dy
Step 4: Distribute the denominator of the first term:
∫(0 to 8) [(y * √y * √2) / 2] - [(y^2)/4] dy
Step 5: Apply the power rule for integration on the first term:
∫(0 to 8) [(√2/2) * y^(3/2)] - [(y^2)/4] dy
Step 6: Integrate each term separately:
[√2/2 * (2/5) * y^(5/2)] - [(1/4) * (1/3) * y^3] from 0 to 8
Step 7: Apply the limits of integration:
[(√2/2 * (2/5) * 8^(5/2)) - (1/4 * (1/3) * 8^3)] - [(√2/2 * (2/5) * 0^(5/2)) - (1/4 * (1/3) * 0^3)]
Step 8: Simplify the expressions with zero:
[(√2/2 * (2/5) * 8^(5/2)) - (1/4 * (1/3) * 8^3)] - [0 - 0]
Step 9: Simplify the expression further:
[(√2 * (8^(5/2)))/5 - (8^3)/12]
Step 10: Rationalize the denominator in the first term:
[(√2 * (8^(5/2)))/5 - (8^3)/12] * (12/12√2)
Step 11: Simplify the expression:
[(12 * √2 * (8^(5/2)))/60 - (12 * (8^3))/12√2]
Step 12: Simplify further:
[√2 * (8^(5/2))/5√2 - 512/√2]
Step 13: Cancel out the terms:
[(8^(5/2))/5 - 512/√2]
Step 14: Simplify the exponent term:
[(32√2)/5 - 512/√2]
Step 15: Rationalize the denominator:
[(32√2)/5 - (512√2)/2]
Step 16: Simplify the expression further:
[(32√2)/5 - 256√2]
Step 17: Find a common denominator:
[(32√2)/5 - (256√2 * 5)/5]
Step 18: Combine the terms:
[(32√2)/5 - (1280√2)/5]
Step 19: Simplify the expression:
[-(1248√2)/5]
Step 20: Multiply the coefficient by 2π:
-(2496π√2)/10
Step 21: Simplify the expression further:
-(1248π√2)/5
So, the correct answer to the integral ∫(0 to 8) [y(√(y/2)-(y/4))]dy is -(1248π√2)/5, which is different from the answer you obtained.