Answer:

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Step-by-step explanation:

To evaluate the given double integral, we can start by integrating with respect to y first and then with respect to x.

First, we integrate with respect to y:

∫(0 to 2/x) e^(xy) dy

Using the power rule for integration, we can integrate e^(xy) with respect to y:

= [1/x * e^(xy)] evaluated from 0 to 2/x

= (1/x * e^(2)) - (1/x * e^(0))

= (1/x * e^(2)) - (1/x * 1)

= (e^(2) - 1)/x

Now, we need to integrate this result with respect to x:

∫(1 to 2) (e^(2) - 1)/x dx

Using the power rule for integration, we can integrate (e^(2) - 1)/x with respect to x:

= [(e^(2) - 1) * ln|x|] evaluated from 1 to 2

= [(e^(2) - 1) * ln|2|] - [(e^(2) - 1) * ln|1|]

= (e^(2) - 1) * ln(2) - (e^(2) - 1) * ln(1)

= (e^(2) - 1) * ln(2)

Therefore, the value of the given integral is (e^(2) - 1) * ln(2).

To reverse the order of integration, we need to rewrite the integral with the limits swapped and change the order of integration:

∫(1 to 2) ∫(0 to 2/x) e^(xy) dy dx

= ∫(0 to 2) ∫(1 to 2/x) e^(xy) dx dy

Now, we can evaluate this reversed integral using the same steps as before.

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Your standard

Answer:

the answer

Step-by-step explanation:

To evaluate the given double integral, we can start by integrating with respect to y first and then with respect to x.

First, we integrate with respect to y:

∫(0 to 2/x) e^(xy) dy

Using the power rule for integration, we can integrate e^(xy) with respect to y:

= [1/x * e^(xy)] evaluated from 0 to 2/x

= (1/x * e^(2)) - (1/x * e^(0))

= (1/x * e^(2)) - (1/x * 1)

= (e^(2) - 1)/x

Now, we need to integrate this result with respect to x:

∫(1 to 2) (e^(2) - 1)/x dx

Using the power rule for integration, we can integrate (e^(2) - 1)/x with respect to x:

= [(e^(2) - 1) * ln|x|] evaluated from 1 to 2

= [(e^(2) - 1) * ln|2|] - [(e^(2) - 1) * ln|1|]

= (e^(2) - 1) * ln(2) - (e^(2) - 1) * ln(1)

= (e^(2) - 1) * ln(2)

Therefore, the value of the given integral is (e^(2) - 1) * ln(2).

To reverse the order of integration, we need to rewrite the integral with the limits swapped and change the order of integration:

∫(1 to 2) ∫(0 to 2/x) e^(xy) dy dx

= ∫(0 to 2) ∫(1 to 2/x) e^(xy) dx dy

Now, we can evaluate this reversed integral using the same steps as before.

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Your standard