Answer:
⁵/₃ ln 2
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