Question

Find the volume of the solid generated by revolving the region in the first quadrant bounded on the left by the circle x² + y2 = 3, on the right by the line x=√3, and above by the line y-√3, revolving about the y-axis.

Answer 1

To find the volume of the solid generated by revolving the region bounded by the circle x² + y² = 3, the line x = √3, and the line y = √3 about the y-axis, we can use the method of cylindrical shells.

Step-by-step explanation:

To find the volume of the solid generated by revolving the region bounded by the circle x² + y² = 3, the line x = √3, and the line y = √3 about the y-axis, we can use the method of cylindrical shells.

First, we need to find the limits of integration. The region is bounded on the left by the circle, so the lower limit is y = -√(3 - x²). The upper limit is y = √3. The limits for x are from 0 to √3, since x is bounded on the right by the line x = √3.

The formula for the volume of a cylindrical shell is given by V = 2πx(y)(dx), where x is the radius of the shell, y is the height of the shell, and dx is the thickness of the shell. In this case, the radius x is given by √(3 - y²), the height y is given by √3 - (-√(3 - x²)), and the thickness dx is dx.

Integrating with respect to x, we get:

V = ∫(0 to √3) [2πx(√3 - (-√(3 - x²)))] dx

Simplifying, we have:

V = ∫(0 to √3) 2πx(√3 + √(3 - x²)) dx

Using the substitution u = 3 - x², we can rewrite the integral as:

V = ∫(0 to 3) 2π(√(3 - u))(√3 + √u) du

Next, we can evaluate the integral and find the volume of the solid.