reverse the order of integration
integration 1-e integration 0-lnx dy dx
To reverse the order of integration, we need to rewrite the given double integral expression by interchanging the order of integration variables.
The given expression is:
∫∫(1 - e^(∫0^ln(x) dy)) dx
Let's reverse the order of integration by converting it into an equivalent double integral with the integration variables interchanged.
Step 1: Determine the limits of integration for the reversed order.
The outer integral (with respect to x) has limits of integration from 0 to 1, as seen from the given expression.
For the inner integral (with respect to y), the limits are obtained by considering the constraints of the original order of integration. The original limits of integration for y are from 0 to ln(x). Therefore, in the reversed order, the limits for y will be from 0 to 1.
Step 2: Rewrite the expression with the reversed order of integration.
The interchanged expression will be:
∫∫(1 - e^(∫0^1 dy)) dx
Step 3: Simplify the inner integral.
The inner integral is with respect to y and has limits from 0 to 1. Since the integral of a constant with respect to y is simply y, we can simplify the inner integral to y evaluated from 0 to 1, which gives:
∫∫(1 - e^(1 - 0)) dx
∫∫(1 - e^1) dx
Step 4: Evaluate the outer integral.
The outer integral is with respect to x and has limits from 0 to 1. Integrating the expression (1 - e^1) with respect to x gives:
∫(x - e^x) evaluated from 0 to 1
[(1 - e^1) - (0 - e^0)]
Step 5: Evaluate the final result.
[(1 - e) - (0 - 1)]
1 - e + 1
2 - e
Therefore, reversing the order of integration results in the final answer of 2 - e.