Integrate[[E^(y/x),{y,0,1},{x,y,1}]
Solution:0.859141 or .5*(e-1)
I can't figure out what to do,and I tried substitution.
To solve the given integral, we can use the technique of changing the order of integration. Here's how you can proceed:
1. Start by writing the given integral:
∫∫E^(y/x) dy dx over the domain y = 0 to 1 and x = y to 1.
2. To simplify the expression, we can rewrite E^(y/x) as E^(y*x^(-1)).
3. Now, let's consider the bounds of integration. Since y varies from 0 to 1 and x varies from y to 1, we can rewrite the integral as:
∫∫E^(y*x^(-1)) dx dy over the domain y = 0 to 1 and x = y to 1.
4. Next, let's interchange the order of integration. To do this, we need to swap the roles of x and y:
∫∫E^(y*x^(-1)) dy dx over the domain x = 0 to 1 and y = x to 1.
5. Now, let's evaluate the inner integral first. Integrating E^(y*x^(-1)) with respect to y will give:
∫[E^(y*x^(-1))] evaluated from y = x to 1.
6. Substituting the upper limit (1) and the lower limit (x) into the expression, we get:
E^(1*x^(-1)) - E^(x*x^(-1)) = E^x - E.
7. Now we can evaluate the outer integral, integrating the result E^x - E with respect to x over the domain x = 0 to 1:
∫[(E^x - E)] evaluated from x = 0 to 1.
8. Substituting the upper limit (1) and the lower limit (0) into the expression, we get:
E^1 - E^0 = E - E^0 = E - 1.
Therefore, the solution to the given integral is E - 1, which is approximately 0.718281 - 1 = -0.281718.
It seems there might be some mistake in the given solution. The value of 0.859141 doesn't match with our calculated value of E-1.