Question

Find the volume of the solid formed by rotating the region enclosed below-axis.
Volume =______.

Answer 1

1. The volume of the solid formed by rotating the region enclosed by
`\(y=e^(2x) + 3\), \(y=0\), \(x=0\), \(x=0.9\)` about the x-axis is approximately
`\(51.67 \pi\).`

2. The volume of the solid obtained by rotating the region bounded by
`\(y=(1)/(√(x+3))\), \(x=3\), \(x=9\), and \(y=0\)` about the x-axis is
`\(\pi \ln 2\)`.

3. For the solid formed by rotating the region bounded by
`\(y=e^(3x)+3\)` and
`\(x=0.5\)` about the x-axis, the volume depends on the upper limit of integration
`\(b\)`.

Question 1:

To find the volume of the solid formed by rotating the region enclosed by
`\(y=e^(2x) + 3\), \(y=0\), \(x=0\), \(x=0.9\)` about the x-axis, we can use the disk method. The volume
`\(V\)` is given by:

`\[V = \pi \int_(0)^(0.9) (e^(2x) + 3)^2 \,dx\]`

Now, integrate with respect to \(x\):

`\[V = \pi \int_(0)^(0.9) (e^(4x) + 6e^(2x) + 9) \,dx\]`

`\[V = \pi \left[(1)/(4)e^(4x) + 3e^(2x) + 9x\right]_(0)^(0.9)\]`

`\[V = \pi \left[(1)/(4)e^(3.6) + 3e^(1.8) + 9(0.9)\right]\]`

`\[V \approx 51.67 \pi\]`

Therefore, the volume of the solid formed by rotating the region about the x-axis is approximately
`\(51.67 \pi\)`.

Question 2:

To find the volume of the solid obtained by rotating the region bounded by
`\(y=(1)/(√(x+3))\), \(x=3\), \(x=9\)`, and
`\(y=0\)` about the x-axis, we again use the disk method. The volume
`\(V\)` is given by:

`\[V = \pi \int_(3)^(9) \left((1)/(√(x+3))\right)^2 \,dx\]`

Now, integrate with respect to
`\(x\):`

`\[V = \pi \int_(3)^(9) (1)/(x+3) \,dx\]`

`\[V = \pi \ln|x+3| \Big|_(3)^(9)\]`

`\[V = \pi (\ln 12 - \ln 6)\]`

`\[V = \pi \ln 2\]`

The volume of the solid formed by rotating the region about the x-axis is
`\(\pi \ln 2\)`.

Question 3:

The solid formed by rotating the region bounded by
`\(y=e^(3x)+3\) and \(x=0.5\)` about the x-axis can be calculated using the disk method:

`\[V = \pi \int_(0.5)^(b) (e^(3x) + 3)^2 \,dx\]`

After integrating and evaluating, the volume
`\(V\)` can be expressed in terms of
`\(b\)`.

The volume of the solid formed by rotating the region about the x-axis depends on the upper limit of integration
`\(b\)`.

The question probable maybe:

1. Find the volume of the solid formed by rotating the region enclosed by y=e^2x +3, y=0, x=0, x=0.9 about the x-axis. Volume:____

2. Find the volume of the solid obtained by rotating the region bounded by y=1/√ (x+3), x=3, x=9, and y=0, about the x-axis. Volume:____

3. A solid is formed by rotating the region in the first quadrant of the xy-plane bounded by the graphs of y=e^3x+3 and x=0.5 about the x-axis. Volume:____