Volume Square Cross 9Dfe10

1. **State the problem:** We need to find the volume of a solid whose base is the region in the first quadrant bounded by the y-axis ($x=0$), the x-axis ($y=0$), the curve $y = e^x$, and the vertical line $x=3$. Each cross section perpendicular to the x-axis is a square. 2. **Formula for volume with square cross sections:** If the cross section perpendicular to the x-axis at position $x$ is a square, then the side length of the square is the height of the region at $x$, which is $y = e^x$. The area of the square cross section at $x$ is: $$A(x) = (e^x)^2 = e^{2x}$$ The volume is the integral of the cross-sectional area along $x$ from $0$ to $3$: $$V = \int_0^3 A(x) \, dx = \int_0^3 e^{2x} \, dx$$ 3. **Evaluate the integral:** $$\int e^{2x} \, dx = \frac{e^{2x}}{2} + C$$ So, $$V = \left[ \frac{e^{2x}}{2} \right]_0^3 = \frac{e^{6}}{2} - \frac{e^{0}}{2} = \frac{e^{6} - 1}{2}$$ 4. **Final answer:** The volume of the solid is: $$\boxed{\frac{e^{6} - 1}{2}}$$