Infinitesimals of Higher Order: a. Figure 11-7f shows a lower Riemann sum for y = mx

Chapter 11, Problem C4

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Infinitesimals of Higher Order: a. Figure 11-7f shows a lower Riemann sum for y = mx, where m 0. The area of each strip is dA y dx, the area of the rectangle. The length of each piece of graph is dL dx. Both approximations become exact as x approaches zero. Show that on integrating from a to b, dA y dx gives the exact area of the region, but dL dx does not give the exact length Figure 11-7fb. Figure 11-7g shows the cone formed byrotating about the x-axis the graph of y = mx (m 0) from x = 0 to x = h. Planesections cut the cone into frustums ofvolume . Each frustum has areadS 2 y dx. Both approximations becomeexact as x approaches zero. Show that onintegrating from 0 to h, givesthe exact volume of the cone, butdS 2 y dx does not give the exactsurface area.Figure 11-7gc. Find the exact area of a strip in Figure 11-7fand the exact volume of a frustum inFigure 11-7g. (The volume of a frustum ofheight dx is V = ( / 3)(R2 + Rr + r2)(dx),where R is the larger radius and r is thesmaller radius of the frustum.)d. A quantity, such as 2 y dx, that approacheszero as x approaches zero is called afirst-order infinitesimal. A quantity, suchas 0.5 y dx, that is the product of two ormore first-order infinitesimals is called ahigher-order infinitesimal. Show thatthe approximations dA y dx and dV y2 dx differ from the exact values inpart c only by infinitesimals of higher order.e. You recall that the differential of arc lengthis dL = . The approximationdL dx = leaves out the first-orderinfinitesimal . Make a conjecture about how accurate a differential of a quantitymust be so that it yields the exact valuewhen it is integrated.f. The second-order infinitesimal 0.5 y dxappears in the exact value of the area ofthe strip in Figure 11-7f in part d. Thelimit of the Riemann sum of such ahigher-order infinitesimal equals zero. Givea reason for each step in this example ofthat statement 0.5 y dx = 0.5 y dx= 0.5 y (ba) 0.5 y dx = 0.5(0)(ba) = 0Property: Infinitesimals of Higher OrderIf dQ Q leaves out only infinitesimalsof higher order, thendQ is exactly equal to Q