https://file.aiquickdraw.com/mathbot/file/1726247850697fm4crntw.jpg

The integral shown in the image is:

\[
\int_0^1 \int_0^x e^x \, dy \, dx
\]

### Steps to solve:

1. **Inner Integral**: First, solve the inner integral with respect to \(y\):
   \[
   \int_0^x e^x \, dy
   \]
   Since \(e^x\) is independent of \(y\), we treat it as a constant:
   \[
   \int_0^x e^x \, dy = e^x \cdot \left[y \right]_0^x = e^x \cdot (x - 0) = x e^x
   \]

2. **Outer Integral**: Now integrate the result with respect to \(x\):
   \[
   \int_0^1 x e^x \, dx
   \]
   To solve this, we use integration by parts:
   - Let \( u = x \), so \( du = dx \).
   - Let \( dv = e^x \, dx \), so \( v = e^x \).

   Using the integration by parts formula:
   \[
   \int u \, dv = uv - \int v \, du
   \]
   we get:
   \[
   \int_0^1 x e^x \, dx = \left[ x e^x \right]_0^1 - \int_0^1 e^x \, dx
   \]
   The first part evaluates to:
   \[
   \left[ x e^x \right]_0^1 = (1 \cdot e^1) - (0 \cdot e^0) = e
   \]
   The second part is:
   \[
   \int_0^1 e^x \, dx = \left[ e^x \right]_0^1 = e - 1
   \]
   Therefore, the solution to the outer integral is:
   \[
   e - (e - 1) = 1
   \]

### Final Answer:
\[
\int_0^1 \int_0^x e^x \, dy \, dx = 1
\]

Would you like further details or clarifications?

Here are 5 related questions to explore:

1. How do you apply integration by parts in more complex cases?
2. What are common uses of double integrals in physics or engineering?
3. How do you evaluate double integrals over different limits?
4. Can double integrals be applied to non-rectangular regions?
5. What are some common applications of the exponential function \( e^x \) in integrals?

**Tip:** When dealing with double integrals, always ensure you understand how the limits of integration interact with the function you're integrating.

Evaluate the double integral \( \int_0^1 \int_0^x e^x \, dy \, dx \).

This page walks through solving the double integral \( \int_0^1 \int_0^x e^x \, dy \, dx \) using integration by parts and properties of exponentials. The step-by-step solution is ideal for students in undergraduate-level calculus.

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