The curves f(x)= 3-x^2 and g(x) =e^(2x)-1 are shown in the figure. Let R be the shaded region bounded by the graph of f (x), g(x) and the y-axis. Find the volume of the solid generated when R is rotated about the x-axis.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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The curves f(x)= 3-x^2 and g(x) =e^(2x)-1 are shown in the figure. Let R be the shaded region bounded by the graph of f (x), g(x) and the y-axis. Find the volume of the solid generated when R is rotated about the x-axis.
The image presents a mathematical graph featuring two curves and a shaded region labeled 'R'. 

### Graph Details:
- **Axes:**
  - **Horizontal Axis (x-axis):** Ranges from -2 to 2 with ticks at intervals of 0.5.
  - **Vertical Axis (y-axis):** Ranges from -1 to 3.5 with ticks at intervals of 0.5.

- **Curves:**
  - The **blue curve** starts from the lower-left corner and moves upward, likely representing an exponential or logarithmic function, crossing the x-axis near the point (0,0).
  - The **brown curve** forms an inverted parabola, peaking above the x-axis at around the midpoint.

- **Shaded Region:**
  - The area marked as 'R' is shaded in red, situated between the two curves. This region is bound by the curves and vertical lines at approximately \( x = 0 \) and \( x = 1.5 \).

### Explanation:
The diagram likely illustrates the integration or area calculation between two curves over a specified interval. The shaded region 'R' is of interest for determining the area between the curves, which may be computed using integral calculus methods for applications in mathematics or physics.
Transcribed Image Text:The image presents a mathematical graph featuring two curves and a shaded region labeled 'R'. ### Graph Details: - **Axes:** - **Horizontal Axis (x-axis):** Ranges from -2 to 2 with ticks at intervals of 0.5. - **Vertical Axis (y-axis):** Ranges from -1 to 3.5 with ticks at intervals of 0.5. - **Curves:** - The **blue curve** starts from the lower-left corner and moves upward, likely representing an exponential or logarithmic function, crossing the x-axis near the point (0,0). - The **brown curve** forms an inverted parabola, peaking above the x-axis at around the midpoint. - **Shaded Region:** - The area marked as 'R' is shaded in red, situated between the two curves. This region is bound by the curves and vertical lines at approximately \( x = 0 \) and \( x = 1.5 \). ### Explanation: The diagram likely illustrates the integration or area calculation between two curves over a specified interval. The shaded region 'R' is of interest for determining the area between the curves, which may be computed using integral calculus methods for applications in mathematics or physics.
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