The volume of the solid obtained by rotating the region enclosed by y=e^(5x)+2 y=0 x=0 x=0.7

about the x-axis can be computed using the method of disks or washers via an integral

A=?
b=?

V=?

using discs, just add up the volumes of tiny slices with radius y and thickness dx:

v = ∫[0,0.7] πy^2 dx
= ∫[0,0.7] π(e^(5x)+2)^2 dx

using shells, you add up the volume of thin cylinders of radius y and height 0.7-x, and thickness dy.
Since the thickness is dy, you have to integrate over y, so you need to express x in terms of y:
x = ln(y-2)/5

Also, for 0<y<2, you have just a solid cylinder of radius 2 and height 0.7

v = π(2^2)(.7) + ∫[2,e^3.5+2] 2πy(0.7-ln(y-2)/5) dy

To find the volume of the solid using the method of disks or washers, we need to set up the integral based on the given information.

The region enclosed by the curves y = e^(5x) + 2, y = 0, x = 0, and x = 0.7 is a finite region between the x-axis and the curve.

First, let's find the limits of integration:

a = 0 (lower limit of x)
b = 0.7 (upper limit of x)

To compute the volume, we will integrate the cross-sectional area of the solid from a to b.

Next, we need to determine the radius of each disk or washer. Since we are rotating about the x-axis, the radius is the distance from the x-axis to the curve at each x-value.

The radius, in this case, is given by r = y.

The differential thickness, dx, will be used to represent the width or thickness of each disk or washer.

The cross-sectional area of each disk or washer, dA, is given by πr^2, where r is the radius.

Therefore, dA = π(y)^2*dx.

To find the volume, we need to integrate dA from a to b:

V = ∫[a to b] π(y)^2*dx

Now, let's substitute the given equation of the curve into the integral:

V = ∫[0 to 0.7] π(e^(5x) + 2)^2*dx

We can now calculate the volume by evaluating this integral.

To find the volume of the solid obtained by rotating the region enclosed by the curves y = e^(5x) + 2, y = 0, and x = 0.7 about the x-axis using the method of disks or washers, we can follow these steps:

Step 1: Sketch the region enclosed by the given curves. This will help visualize the shape and understand the limits of integration.

Step 2: Determine the range of x-values over which the region is enclosed. In this case, we are given that x ranges from 0 to 0.7.

Step 3: Identify the function that represents the outer radius (R(x)) and the function that represents the inner radius (r(x)) of the solid.

In this case, the outer radius (R(x)) is the distance between the x-axis and the curve y = e^(5x) + 2, and the inner radius (r(x)) is the distance between the x-axis and the x = 0 line (since y = 0).

Step 4: Express the outer radius (R(x)) and inner radius (r(x)) as functions of x.

- The outer radius (R(x)) for this problem is R(x) = e^(5x) + 2.
- The inner radius (r(x)) is r(x) = 0.

Step 5: Write the formula for the volume of a solid using the method of disks or washers. The formula for a disk is:

dV = π[R(x)^2 - r(x)^2] dx

Step 6: Integrate the above expression with respect to x over the given range. The integral formula to find the volume (V) is:

V = ∫[a to b] π[R(x)^2 - r(x)^2] dx

In this case, the limits of integration are from x = 0 to x = 0.7.

So, the final expression will be:

V = ∫[0 to 0.7] π[(e^(5x) + 2)^2 - 0] dx

Now, integrating this expression will give you the volume (V) of the solid.

To compute the volume (V), you can evaluate the definite integral using numerical methods or approximate it using software or calculators capable of integrating functions.

Therefore,

A = [0, 0.7]
b = 0
V = ∫[0 to 0.7] π[(e^(5x) + 2)^2 - 0] dx

Please note that numerical values of A and b cannot be determined without further context or information.