Find the volume (using cylindrical disks) of the figure obtained by rotating the

Question

Explanation / Answer

First solve for points of intersection to find the bounds of the bounded curve: ?x = x2 => x3/2 = 0 has solutions x = 0 and x = 1 The curve formed from x = 0 to x = 1 is bounded below by y = x2 and above by y = ?x Note: these curves are symmetrical about the live y = x, so you can choose to revolve the shape around either the x or y axis, the answer will be the same. Find volume via slicing, aka disc method: In general, to find the volume of a curve rotated around the y-axis: dV = ?r2dy => ?(f(y))2dy 1) V = Since there are two curves, the volume found by revolving the curve closest to the axis must be subtracted from the volume found by revolving the curve furthest from the axis. The problem gives the curves as functions of x, but eqn. 1) requires functions of y: f(x) = ?x => f1(y) = y2 f(x) = x2 => f2(y) = ?y The curve furthest from the y-axis is f2, and the closest is f1 So, using eqn. 1) with y1 = 0 and y2 = 1: V = V(f2) - V(f1)Finding colume via cylindrical shells: In general, to find the volume of a curve rotated around the y-axis: 2) The volume closest to the x-axis must be subtracted from that of the furthest curve. In this case the closest to the x-axis is f1(x) = x2, and the furthest f2(x) = ?x So using eqn. 2) with x1 = 0 and x2 = 1: V = V(f2) - V(f1)

Related Questions

Navigate

• Browse (All)
• Browse F
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.