2. [-/0.13 Points] DETAILS SCALCET9 6.2.AE.008. 0/100 Submissions Used MY NOTES ASK YOUR TEACHER Example 8 Video Example () Find the volume of a pyramid whose base is a square with side _ and whose height is h. Solution We place the origin O at the vertex of the pyramid and the x-axis along its central axis as in the following figure. h O Any plane P that passes through x and is perpendicular to the x-axis intersects the pyramid in a square with side of length s. We can express s in terms of x by observing from the similar triangles in the figure below that _ = N| ~ NIK -, and so h S = O Another method is to observe that the line OP has slope L (2h) ( 2h ) ' ] - and so its equation is y = _. Thus, the cross-sectional area is A ( x ) = 52 = The pyramid lies between x = 0 and x = so its volume is V = " A(x) ax = ["12 x2 ax = Need Help? Read It4:19 Search webassign.net Your last submission is used for your score. 1. [-/0.13 Points] DETAILS SCALCET9 6.2.AE.004. 0/100 Submissions Used MY NOTES ASK YOUR TEACHER Example 4 The region & enclosed by the curves y = 3x and y = x2 is rotated about the x-axis. Find the volume of the resulting solid. Solution The curves y = 3x and y = x2 intersect at the points (0, and (3, . The region between them, the solid of rotation, and a cross section perpendicular to the x-axis are shown in the figures below. y = 3x y = x2 Graph Graph Description - - NWAVIO VOODO -3 -2 -1 A(x) 3 x A cross-section in the plane PX has the shape of a washer (an annular ring) with inner radius and an outer radius so we find the cross-sectional area by subtracting the area of the inner circle from the area of the outer circle. A(x) = 9xx2 - x(x2)2 = Therefore, we have v = / A(x ) ax = [ *(9x 2 - * * ) dx Need Help? Read It