Evaluate iterated integral, ∫(2,1) ∫(x^3,x) e^(y/x) dydx
AAAaannndd the bot gets it wrong yet again!
∫[1..2] ∫[x..x^3] e^(y/x) dy dx
= ∫[1..2] x(e^(x^2)-e) dx
= 1/2 (e^x^2 - ex^2) [1..2]
= e/2 (e^3 - 4)
I apologize for the error in my previous response. Thank you for providing the correct solution. You are correct that the iterated integral evaluates to:
∫[1..2] ∫[x..x^3] e^(y/x) dy dx
= ∫[1..2] x(e^(x^2)-e) dx
= 1/2 (e^x^2 - ex^2) [1..2]
= e/2 (e^3 - 4)
To evaluate the iterated integral ∫(2,1) ∫(x^3,x) e^(y/x) dy dx, we will start by evaluating the inner integral with respect to y, and then substitute the result into the outer integral.
∫(2,1) ∫(x^3,x) e^(y/x) dy dx
Step 1: Evaluate the inner integral with respect to y:
∫(x^3,x) e^(y/x) dy
Using the power rule for integration, we can rewrite the integral as:
∫(x^3,x) e^(y/x) dy = [ x * e^(y/x) ] evaluated from y = x^3 to y = x
= x * (e^(x/x) - e^(x^3/x))
Simplifying further, we have:
= x * (e^1 - e^(x^2))
Step 2: Substitute the result into the outer integral:
∫(2,1) (x * (e^1 - e^(x^2))) dx
Now we integrate with respect to x:
∫(2,1) (x * (e^1 - e^(x^2))) dx = [ (1/2) * x^2 * (e^1 - e^(x^2)) ] evaluated from x = 1 to x = 2
= (1/2) * (2^2) * (e^1 - e^(2^2)) - (1/2) * (1^2) * (e^1 - e^(1^2))
= 2 * (e^1 - e^4) - (1/2) * (e - 1)
Simplifying further, we have:
= 2e - 2e^4 - (1/2)e + 1/2
= -2e^4 + (3/2)e - 1/2
Therefore, the value of the iterated integral is -2e^4 + (3/2)e - 1/2.
We can start by evaluating the inner integral first with respect to y while treating x as a constant:
∫(2,1) e^(y/x) dy evaluated at y=x^3 and y=x
= ∫(2,1) e^(y/x) dx^3 - ∫(2,1) e^(y/x) dx
= x^3(e - e^(8/x))/x - (e - e^(2/x))
= x^2(e - e^(8/x)) - (e - e^(2/x))
Next, we can integrate this expression with respect to x:
∫(1,2) x^2(e - e^(8/x)) - (e - e^(2/x)) dx
= [1/3 x^3 (e - e^(8/x)) - x(e - e^(2/x))] evaluated at x=1 and x=2
= [1/3 (2^3 - 1^3)(e - e^(8/2)) - 2(e - e^(2/2))] - [1/3 (1^3 - 1^3) (e - e^(8/1)) - 1(e - e^(2/1))]
= 2/3(e - e^4) - 2e + 2 - 1/3(e - e^8) + 1(e - 1)
= 2/3 - 1/3 e^8 + 8/3 e^4 - 2e + 1
= -2/3 + 8/3 e^4 - 1/3 e^8
Therefore, the final answer is -2/3 + 8/3 e^4 - 1/3 e^8.