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Answer As we can see in the gure, the line y= 2x 7 lies above the parabola y= x2 1 in the region we care about26/5/ In this section we will start evaluating double integrals over general regions, ie regions that aren't rectangles We will illustrate how a double integral of a function can be interpreted as the net volume of the solid between the surface given by the function and the xyplaneSet y y equal to the new right side y = x 2 y = x 2 y = x 2 y = x 2 Use the vertex form, y = a ( x − h) 2 k y = a ( x h) 2 k, to determine the values of a a, h h, and k k a = 1 a = 1 h = 0 h = 0 k = 0 k = 0 Since the value of a a is positive, the parabola opens up Opens Up




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Assignment 7 (MATH 215, Q1) 1 Find the area of the given surface (a) The part of the cone z = p x2 y2 below the plane z = 3 Solution The surface can be represented by the vector equationSOLUTIONS TO MAXIMUM/MINIMUM PROBLEMS SOLUTION 1 Let variables x and y represent two nonnegative numbers The sum of the two numbers is given to be 9 = x y , so that y = 9 x We wish to MAXIMIZE the PRODUCT P = x y2 However, before we differentiate the righthand side, we will write it as a function of x onlyTo find the answer, make a data table Data Table for y = x 2 And graph the points, connecting them with a smooth curve Graph of y = x 2 The shape of this graph is a parabola Note that the parabola does not have a constant




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Math 2260 Exam #1 Practice Problem Solutions 1What is the area bounded by the curves y= x2 1 and y= 2x 7?Over the region D = {(x,y) x2 y2 8} As before, we will find the critical points of f over DThen,we'llrestrictf to the boundary of D and find all extreme values It is in this second step that we will use Lagrange multipliers The region D is a circle of radius 2 p 2Find the area bounded by the curve y = x2x4, the xaxis and the ordinates x = 1 and x = 3 Solution If we set y = 0 we obtain the quadratic equation x2 x 4 = 0, and for this quadratic b2 − 4ac = 1− 16 = −15 so that there are no real roots This means that the curve does not cross the x




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The earliest known work on conic sections was by Menaechmus in the 4th century BC He discovered a way to solve the problem of doubling the cube using parabolas (The solution, however, does not meet the requirements of compassandstraightedge construction)The area enclosed by a parabola and a line segment, the socalled "parabola segment", was computed byWolframAlpha brings expertlevel knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels23/9/11 13–86 A 2kg particle travels along a horizontal smooth path defined by 1 t2 r = ¢ t3 2 ≤ m, u = ¢ ≤ rad, 4 4 where t is in seconds Determine the radial and transverse components of




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