Final answer:
To find the volume of the solid generated by revolving the triangular region about the x-axis, we can use the disk method. The volume can be found by integrating the cross-sectional area of each infinitesimally thin disk, which can be calculated using the formula A=π(y)². The limits of integration are determined by the intersection points of the curve y=4/x³ and the line x=1.
Step-by-step explanation:
To find the volume of the solid generated by revolving the triangular region about the x-axis, we can use the disk method. First, let's find the limits of integration by setting up the inequality 1 ≤ x ≤ 2 (the intersection points of the curve y=4/x³ and the line x=1). Now, let's find the cross-sectional area of each infinitesimally thin disk by taking a cross-section parallel to the x-axis at a distance x and find the radius of the disk at that point, which is y=4/x³. The volume of the solid can be found by integrating the cross-sectional area from x=1 to x=2.
The cross-sectional area of each disk is given by A=π(y)², where y=4/x³. Substituting y=4/x³, we get A=π(4/x³)²=π(16/x⁶). The volume of the solid can be found by integrating the cross-sectional area with respect to x: V=∫12π(16/x⁶)dx=π∫12(16/x⁶)dx. Solving this integral will give us the volume of the solid.