Question

Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line.

1. y= In x, y=0, x=2; about the y-axis
2. y= sin^-1 x , y= pie/2, x=0; about y=3

Answer 1

To find the volume of the solid obtained by rotating the region bounded by the given curves about specified lines, we use the method of cylindrical shells. The formula for the volume of the solid obtained by rotating the region about the y-axis is V = ∫πr(x)h(x)dx, where r(x) is the distance from the axis of rotation to the function and h(x) is the height of each shell. For the given curves y = ln x, y = 0, and x = 2, the integral would be V = ∫πx(ln x)dx. Similarly, for the given curves y = sin^-1 x, y = π/2, and x = 0, and the line y = 3, the integral would be V = ∫π(3 - y)(x)dy.

Step-by-step explanation:

To set up the integral for the volume of the solid obtained by rotating the region bounded by the curves y = ln x, y = 0, and x = 2 about the y-axis, we will use the method of cylindrical shells. The formula for the volume of a solid obtained by rotating the region bounded by the curves about the y-axis is:

V = ∫πr(x)h(x)dx

where r(x) is the distance from the axis of rotation to the function, and h(x) is the height of each shell. In this case, r(x) = x and h(x) = y. Plugging these values into the formula, we get:

V = ∫πx(ln x)dx

To set up the integral for the volume of the solid obtained by rotating the region bounded by the curves y = sin^-1 x, y = π/2, and x = 0 about the line y = 3, we will also use the method of cylindrical shells. The formula for the volume of a solid obtained by rotating the region bounded by the curves about a vertical line is:

V = ∫πr(y)h(y)dy

where r(y) is the distance from the axis of rotation to the function, and h(y) is the height of each shell. In this case, r(y) = 3 - y and h(y) = x. Plugging these values into the formula, we get:

V = ∫π(3 - y)(x)dy

Answer 2

To find the volume of the solid obtained by rotating the given regions about the specified lines, use the method of shells for the first case and the method of washers for the second case, setting up but not evaluating the respective integrals.

Step-by-step explanation:

Volume of Solids of Revolution

To set up the integral for the volume of the solid obtained by rotating the region bounded by the curves y = ln(x), y = 0, and x = 2 about the y-axis, we use the method of shells, which involves the use of cylindrical shells. For the first case, the volume V is given by the integral:

V = 2π ∫₀² x ln(x) dx

In the second scenario, where the region bounded by the curves y = sin-1(x), y = π/2, and x = 0 is rotated about the line y = 3, we use the method of washers. The appropriate integral for this volume would be:

V = π ∫₀¹ (³ - (3 - sin-1(x)))2 dx

Both integrals represent the volume of the solid of revolution and are set up according to the given parameters but not evaluated.