Determine the volume of the solid formed by rotation about the y-axis of the region bounded by the curves y=4^x-1 and y = 21x on the interval 0< x<3.

Verify this equation, y = 4^x-1

You need parentheses.

Is it, y = 4^(x - 1)?
Or, y = 4^(x) - 1?
Or ?

To determine the volume of the solid formed by rotating the region bounded by the curves about the y-axis, you can use the method of cylindrical shells.

Step 1: Sketch the region bounded by the curves to visualize it. The curves are y = 4^x - 1 and y = 21x. On the interval 0 < x < 3, the region is bounded between these two curves.

Step 2: Determine the height of the cylindrical shells. Since we are rotating the region about the y-axis, the height of each shell will be the difference between the two curves at a particular value of y. The difference between the curves can be calculated as (4^x - 1) - (21x).

Step 3: Determine the radius of the cylindrical shells. The radius of each shell will be the value of x. Since the shells are parallel to the y-axis, the radius remains constant.

Step 4: Determine the bounds of integration. In this case, we integrate with respect to y, so we need to determine the range of y-values that correspond to the region. To find the lower and upper bounds of y, set the two curves equal to each other and solve for x:
4^x - 1 = 21x

This equation cannot be solved algebraically, so we'll have to use approximation methods or numerical methods to find the x-values where the curves intersect. However, for the purpose of explanation, let's assume that the curves intersect at x = a and x = b, where a and b are within the interval 0 < x < 3.

Step 5: Set up the integral for the volume. We integrate the product of the height and circumference of each shell over the range of y. The volume can be calculated using the formula:

V = ∫[a to b] 2πx(4^x - 1 - 21x) dy

Step 6: Evaluate the integral. Once you have set up the integral, you can evaluate it using integration techniques or numerical approximation methods such as a calculator, software, or computer program.

Note: Since the exact intersection points of the curves were not provided, it is not possible to provide the exact values for the bounds of integration and the solution. However, by following these steps, you can find the volume of the solid formed by rotation about the y-axis for any given intersection points between the curves y = 4^x - 1 and y = 21x on the interval 0 < x < 3.