calcuate the double integral(e^(x/y))dydx for [sqrt(x),1]x[0,1]

We are tasked with evaluating the double integral:

\[
\int_0^1 \int_{\sqrt{x}}^1 e^{\frac{x}{y}} \, dy \, dx
\]

### Step-by-step solution:

1. **Inner Integral (with respect to \( y \)):**
   \[
   I_1 = \int_{\sqrt{x}}^1 e^{\frac{x}{y}} \, dy
   \]

   To solve this, let's use the substitution method:
   - Let \( u = \frac{x}{y} \), which gives \( du = -\frac{x}{y^2} dy \), and rearranging gives \( dy = -\frac{y^2}{x} du \).

   Substituting the bounds:
   - When \( y = \sqrt{x} \), \( u = \frac{x}{\sqrt{x}} = \sqrt{x} \).
   - When \( y = 1 \), \( u = x \).

2. **Outer Integral (with respect to \( x \)):**

   After finding the result of the inner integral, we can integrate with respect to \( x \) over the interval \([0, 1]\).

Let me proceed to calculate the double integral.The value of the double integral

\[
\int_0^1 \int_{\sqrt{x}}^1 e^{\frac{x}{y}} \, dy \, dx
\]

is \( \frac{1}{2} \).

Would you like further details or have any questions?

Here are some related problems to explore:
1. Evaluate \(\int_0^1 \int_0^x e^{x+y} \, dy \, dx\).
2. Find the double integral \(\int_0^1 \int_0^x y^2 \, dy \, dx\).
3. Compute \(\int_0^1 \int_0^1 xy \, dy \, dx\).
4. Solve \(\int_0^2 \int_0^x \sin(y) \, dy \, dx\).
5. Determine \(\int_0^1 \int_0^1 e^{x+y} \, dy \, dx\).

**Tip:** When dealing with nested integrals, always integrate step-by-step, starting from the inner integral.

Calculate the double integral (e^(x/y)) dy dx for [sqrt(x),1] x [0,1]

This tutorial explains how to evaluate the double integral \( \int_0^1 \int_{\sqrt{x}}^1 e^{\frac{x}{y}} \, dy \, dx \) using the substitution method. Learn step-by-step techniques for solving integrals involving exponential functions and nested limits. Suitable for college-level calculus students.

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