Interchange the order of integration and evaluate the integral [0,8]∫ [∛y,2]∫ (e^x^4)dxdy
x = ∛y, right? y = x^3
This is similar to another problem posted recently -- same region of integration.
Draw a diagram. The region is below the curve y = x^3. So,
∫[0,8] ∫[[∛y,2] f(x,y) dx dy = ∫[0,2] ∫[0,x^3] f(x,y) dy dx
how did you get the x^3
To interchange the order of integration, we need to rewrite the integral with respect to the other variable and change the limits accordingly. Here's how:
The given integral is:
[0,8]∫ [∛y,2]∫ (e^(x^4)) dxdy
Let's first integrate with respect to x.
The inner integral becomes:
∫ (e^(x^4)) dx
To solve this integral, we can use a substitution.
Let u = x^4, then du = 4x^3 dx
Rearranging the equation, dx = du / (4x^3)
Substituting these values in the integral, we get:
∫ (e^u) (du / (4x^3))
Since we want to evaluate the integral with respect to x, we need to express x in terms of u.
From the substitution u = x^4, we can rewrite it as x = u^(1/4).
Substituting this back into the integral, we have:
∫ (e^u) (du / (4(u^(3/4))^3))
Simplifying the expression, we get:
∫ (e^u) / (4u^(9/4)) du
Now we need to calculate the antiderivative of (e^u) / (4u^(9/4)) with respect to u.
Unfortunately, there is no simple closed-form expression for this antiderivative. So, we need to use numerical methods or software to solve it.
Once we have the antiderivative, let's call it F(u), the next step is to evaluate this antiderivative over the x limits, which are ∛y to 2.
The final step is to integrate this resulting expression with respect to y, using the limits [0, 8]. This will yield the final numerical value of the integral.
Therefore, to evaluate the given integral, you need to calculate the antiderivative (F(u)) of (e^u) / (4u^(9/4)) and then perform the double integration with the given limits.