Tiêu đề 77 Volumes using Integration.pdf ✅

MSTC 76: Volumes using Integration 1. Choice of Differential Strip When solving values derived from the bounded area, the choice of differential strips can make the procedure much easier or much harder. A good differential strip is one in which endpoints do not both touch the curve. Consider the area bounded by the parabola and the x-axis, A vertical differential strip is feasible since its top end is on the parabola, and the bottom end is on the x- axis. A horizontal differential strip is impractical since its sides are both on the parabola. 2. Disk Method A solid of revolution is generated by rotating a bounded area about an axis of revolution. The disk method is best used when the differential strip selected touches the axis and is perpendicular to that axis. From the figure, the length of the differential strip becomes the radius of the cylindrical disk, and its width becomes the altitude. From the volume of a cylinder (see MSTC 52: Prisms and Cylinder), V = πr 2h Using a vertical strip, r = y h = dx V = ∫ πy 2dx
Using a horizontal strip, 3. Ring Method The ring method is best used when the differential strip selected is perpendicular to the axis of revolution but does not touch the axis of revolution. It is derived using two disk methods, where the total volume is equal to the outer volume (the curve further from the axis) minus the inner volume (the curve nearer to the axis). Using a vertical strip, Using a horizontal strip, r = x h = dy V = ∫ πx 2dy V = Vouter − Vinner V = ∫ πyo 2dx − ∫ πyi 2 dx V = ∫ π(yo 2 − yi 2 )dx V = ∫ πxo 2dy − ∫ πxi 2 dy V = ∫ π(xo 2 − xi 2 )dx