Evaluate the integral by reversing the order of integration.
(2 integrals)
0 to 8
cube root(y) to 2 8e^(x^4) dx dy
To evaluate the given double integral by reversing the order of integration, we need to switch the order of integration and write it as an iterated integral.
The original integral is: ∫∫ 8e^(x^4) dx dy, with the limits of integration 0 ≤ y ≤ 8 and cube root(y) ≤ x ≤ 2.
Let's reverse the order of integration by integrating with respect to x first.
First, we integrate with respect to x: ∫ 8e^(x^4) dx
To integrate this function, we can use a substitution:
Let u = x^4, then du = 4x^3 dx, and dx = du / (4x^3).
So the integral becomes: ∫ 8e^u (du / (4x^3))
Simplifying: 2e^u/x^3
Now let's evaluate the integral with respect to x, while treating y as a constant:
∫ (2e^u) / x^3 dx
= (2 / y^(3/4)) ∫ e^u dx (using the fact that u = x^4, hence x = u^(1/4))
= (2 / y^(3/4)) e^u + C
Now, let's proceed with integrating this result with respect to y, while treating x as a constant:
∫ (2 / y^(3/4)) e^u dy
= (2 / y^(3/4)) e^u y + C
= 2e^(x^4) y^(1/4) + C
To find the final result of the integral, we evaluate this result at the limits of integration.
The limits for y are from cube root(y) to 2, and for x, they are from 0 to 8.
Therefore, the final answer is:
∫∫ 8e^(x^4) dx dy = ∫(0 to 8) ∫(cube root(y) to 2) 2e^(x^4) y^(1/4) dy dx.
To evaluate the given double integral by reversing the order of integration, follow these steps:
Step 1: Draw the region of integration in the xy-plane.
Start by visualizing the region of integration for the given limits: 0 to 8 for x and cube root(y) to 2 for y. Sketch the rectangular region bounded by the lines x = 0, x = 8, y = cube root(y), and y = 2.
Step 2: Reverse the order of integration.
To reverse the order of integration, we need to express the limits of integration in terms of the other variable. In this case, we are switching from integrating with respect to x first to integrating with respect to y first.
The inner integral will now be with respect to x, so the limits for x will be determined by the intersection points of the horizontal lines (y = constant) with the vertical lines (x = 0 and x = 8).
The outer integral will now be with respect to y, so the limits for y will be determined by the intersection points of the vertical lines (x = constant) with the curves (y = cube root(y) and y = 2).
Step 3: Determine the new limits of integration.
To find the new limits of integration, we set up the integral as follows:
∫∫(Region) 8e^(x^4) dx dy
The region is defined as follows:
0 ≤ x ≤ 8
cube root(y) ≤ y ≤ 2
Step 4: Evaluate the integral by applying the new limits.
Integrating the given function 8e^(x^4) with respect to x, we obtain:
∫(cube root(y) to 2) 8e^(x^4) dx
Integration with respect to x gives:
(2/3)e^(x^4) from 0 to 8
Evaluating this expression gives:
(2/3)e^(8^4) - (2/3)e^(0^4)
Simplifying:
(2/3)e^(4096) - (2/3)
Step 5: Apply the outer integral with the new limits.
Now, we apply the outer integral with respect to y to the result obtained in Step 4:
∫((2/3)e^(4096) - (2/3)) dy
Integration with respect to y gives:
((2y/3)e^(4096) - (2y/3)) from cube root(y) to 2
Evaluating this expression gives:
((2(2)/3)e^(4096) - (2(2)/3)) - ((2(cube root(y))/3)e^(4096) - (2(cube root(y))/3))
Simplifying the expression:
(4/3)e^(4096) - 4/3 - (2/3)e^(4096) + (2/3)cube root(y)
Final Answer:
(2/3)e^(4096) - (4/3) - (2/3)e^(4096) + (2/3)cube root(y)
sketch the region. It will be clear that
∫[0,8] ∫[∛y,2] 8e^x^4 dx dy
= ∫[0,2] ∫[0,x^3] 8e^x^4 dy dx
= ∫[0,2] 8x^3 e^x^4 dx
= 2e^x^4 [0,2]
= 2(e^16-1)
Nice trick, since e^x^4 does not integrate well.