Evaluate the integral by reversing the order of integration.(2 integrals)
0 to 4 5ex2 dx dy
3y to 12
Not sure just what you intended. I (sort of) see
∫[4,12]∫[0,3y] 5e^(x^2) dx dy
It seems wrong; maybe you could fix it so it is
(a) correct
(b) readable
To evaluate the integral by reversing the order of integration, we need to express the given double integral in terms of the opposite order.
The given double integral is:
∫[0 to 4] ∫[3y to 12] 5e^(2x) dx dy
To reverse the order of integration, we need to express the limits of integration in terms of the opposite variables. In this case, we need to express the limits of integration in terms of x rather than y.
Let's start by looking at the limits of integration for the y variable:
The limits of integration for y are 3y to 12.
To determine the new limits of integration for x, we need to solve for y in terms of x. From the first equation, we have:
3y = x
Solving this equation for y, we find:
y = x/3
Now, let's consider the limits of integration for x:
The limits of integration for x are 0 to 4.
We will express these limits in terms of y using the relation we found above:
When x = 0, y = 0/3 = 0.
When x = 4, y = 4/3.
Therefore, the new limits of integration for y are from 0 to 4/3.
Now, we can re-write the double integral with the reversed order of integration:
∫[0 to 4/3] ∫[0 to 5ex/3] 5e^(2x) dy dx
To evaluate the integral by reversing the order of integration, we need to rewrite the given double integral by swapping the order of integration. Here's how you can do it step by step:
Step 1: Write down the original integral:
∫∫(0 to 4) (3y to 12) 5e^(2x) dx dy
Step 2: Reverse the order of integration:
∫∫(3y to 12) (0 to 4) 5e^(2x) dx dy
Step 3: Rewrite the limits of integration:
∫∫(0 to 4) ∫(3y to 12) 5e^(2x) dy dx
Step 4: Evaluate the inner integral with respect to y:
∫∫(0 to 4) [∫(3y to 12) 5e^(2x) dy] dx
Step 5: Integrate the inner integral:
∫∫(0 to 4) [5e^(2x) * y] from 3y to 12 dx
Step 6: Simplify the limits of integration:
∫∫(0 to 4) [5e^(2x) * (12 - 3y)] dx
Step 7: Integrate the outer integral:
∫[∫(0 to 4) [5e^(2x) * (12 - 3y)] dx] dy
At this point, you can continue evaluating the integral by integrating with respect to x.