find the volume of the solid formed by rotating the region enclosed by y=e^(1x+4), y=0, x=0, x=0.9

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To find the volume of the solid formed by rotating the region enclosed by the given curves, we can use the method of cylindrical shells. Here are the step-by-step instructions to calculate it:

1. Begin by graphing the region enclosed by the given curves. In this case, the region is bound by the y-axis (y=0), the x-axis (x=0), the curve y=e^(1x+4), and the vertical line x=0.9.

2. Calculate the height of the region. In this case, the height is given by the difference between the upper and lower curves. Since the lower curve is the x-axis (y=0) and the upper curve is y=e^(1x+4), the height is given by y=e^(1x+4) - 0 = e^(1x+4).

3. Determine the range of x-values that define the region. In this case, the region is bound by the vertical lines x=0 and x=0.9.

4. Set up the integral for the volume using the formula for the cylindrical shells method:

V = ∫(2πrh) dx

where r is the distance from the axis of rotation to the shell, and h is the height of the shell (which we calculated in step 2).

5. Substitute the values into the integral:

V = ∫(2πx * e^(1x+4)) dx

6. Evaluate the integral over the given range of x-values. The integral should be taken from x=0 to x=0.9:

V = ∫[0.9 to 0] (2πx * e^(1x+4)) dx

7. Use a numerical method, such as a calculator or computer program, to evaluate the integral and find the final volume.

By following these steps, you should be able to find the volume of the solid formed by rotating the given region.