Question

set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region in the first quadrant bounded by the curves y = tan2 x, y = 3, and x = 0 about the line y = 3.

Answer 1

To set up the integral for the volume of the solid obtained by rotating the region in the first quadrant bounded by the curves y = tan²x, y = 3, and x = 0 about the line y = 3, we need to determine the bounds of integration and the integrand. The bounds of integration for x will be from 0 to a value that satisfies y = tan^2 x = 3. The integrand will be the cross-sectional area of the solid at a given x-value, which can be determined by subtracting the area of the circle with radius y = 3 from the area under the curve y = tan² x. Therefore, the integral for the volume of the solid is ∫(A - πr²) dx.

Step-by-step explanation:

To set up the integral for the volume of the solid obtained by rotating the region in the first quadrant bounded by the curves y = tan2 x, y = 3, and x = 0 about the line y = 3, we need to first determine the bounds of integration and the integrand.

The region in the first quadrant bounded by the curves can be visualized as a sector of a cone, with the axis of rotation being the line y = 3. We can see that the bounds of integration for x will be from 0 to a value that satisfies y = tan2 x = 3, which can be found by solving the equation tan2 x = 3 for x.

The integrand will be the cross-sectional area of the solid at a given x-value, which can be determined by subtracting the area of the circle with radius y = 3 from the area under the curve y = tan2 x. Therefore, the integral for the volume of the solid is:

∫(A - πr2) dx

Answer 2

To find the volume of the solid obtained by rotating a region bounded by curves, we can use the method of cylindrical shells. The integral for the volume is ∫[0,a] 2πx(3-tan^2(x)) dx.

Step-by-step explanation:

To set up the integral for the volume of the solid, we will use the method of cylindrical shells. The region bounded by the curves y = tan^2(x), y = 3, and x = 0 in the first quadrant will be rotated about the line y = 3. To set up the integral, we will integrate the surface area of the cylindrical shells over the range of x values where the curves intersect.

The height of each cylindrical shell is given by the difference between the line y = 3 and the function y = tan^2(x). The radius of each shell is equal to the value of x. The surface area of each shell can be calculated using the formula 2*pi*r*h.

Therefore, the integral for the volume of the solid is:

V = ∫[0,a] 2πx(3-tan^2(x)) dx, where a is the x-value where the curves intersect.