Evaluate the integral by reversing the order of integration. Integral from 0 to 1 integral from √x to 1 of 5/(y³+1) dydx
1 answer:
Answer:
⁵/₃ ln 2
Step-by-step explanation:
∫₀¹ ∫√ₓ¹ 5 / (y³ + 1) dy dx
We want to change the order of integration. To do this, we start by graphing the region contained by the four limits.
x = 0, x = 1
y = √x, y = 1
(Notice that y = √x is the same as x = y²).
Once we've graphed the region, we need to write the domain of x in terms of y. In this case, 0 < x < y².
Then, we find the range: 0 < y < 1.
Now we can rewrite the integral:
∫₀¹ ∫₀ʸ² 5 / (y³ + 1) dx dy
Notice that the integrand itself, 5 / (y³ + 1), does not change. Only the limits have changed.
Solve the integral.
∫₀¹ [ 5 / (y³ + 1) x |₀ʸ² ] dy
∫₀¹ [ 5y² / (y³ + 1) ] dy
⁵/₃ ∫₀¹ [ 3y² / (y³ + 1) ] dy
(⁵/₃ ln|y³ + 1|) |₀¹
⁵/₃ ln|1³ + 1| − ⁵/₃ ln|0³ + 1|
⁵/₃ ln 2
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Water quantity = 3 parts .
Ratio of Strawberry juice to Water is = 5 : 3.
Total fluid ounces of the strawberry drink need to make = 64 ounces.
Let us assume strawberry juice need = x ounces.
So, the water would be need = Total fluid - strawberry juice = 64 -x ounces.
The ratio of Strawberry juice to Water would reamain same 5:3.
So, we can setup a proportion now.
On cross multiplying, we get
x*3=5 *(64-x)
Distibuting 5 over (64-x), we get
3x = 5 * 64 - 5*x
3x = 320 -5x
Adding 5x on both sides, we get
3x+5x = 320-5x +5x
8x = 320
Dividing both sides by 8, we get
x = 40
So, strawberry juice need = 40 ounces.
Water would be need = 64 - 40 = 24 ounces.
Therefore, 40 ounces of strawberry juice and 24 ounces of water carmen needs to make 64 fluid ounces of the strawberry drink.
The first digit is one of {1, 3, 5, 7, 9}, and the last digit is one of {1, 2, 3, 4, 5}.
If the first digit is one of {1, 3, 5} (3 choices), then the last digit is one of {1, 2, 3, 4, 5} minus whatever is picked for the first digit (4 choices, and the remaining eight digits can be arranged in 8! ways, so there are 12*8! possible permutations.
If the first digit is one of {7, 9} (2 choices, then the last digit is one of {1, 2, 3, 4, 5} (5 choices), and again there are 8! ways of arranging the remaining digits, so there are 10*8! possible permutations.
Then the total number of permutations that fit the criteria is 12*8! + 10*8! = 22*8!. There are 10! total permutations that can be made overall, so the probability of randomly picking one we want is