Long description: fig1
Alt text: The image displays the mathematical integration of the exponential function \(e^{-x}\) from \(0\) to \(\text{infinity}\), alongside its graph.
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- The left side of the image shows the step-by-step calculation of the definite integral \(\int_{0}^{\infty} e^{-x} \text{d}x\).
- The first step rewrites the integral as \(\left[-e^{-x}\right]_{0}^{\infty}\).
- The second step evaluates the limits, resulting in \((-e^{-(\infty)}) - (-e^{-0})\).
- The third step simplifies the expression to \((0) - (-e^{-0})\).
- The final step calculates the result, yielding \(1\).
- The right side of the image is a graph plotting the function \(y = e^{-x}\) against \(x\).
- The curve starts at the point \((0, 1)\) on the vertical axis and decreases rapidly, approaching the horizontal axis (the \(x\)-axis) as \(x\) increases towards positive infinity.
- The horizontal axis is labeled with \(x\), and the vertical axis is labeled with \(e^{-x}\).