Applied Mathematics 1413 Chapter 7.1-7.2: Applied Mathematics 1413 Chapter 7.1-7.: Applied Mathematics 1413 Chapter 7.1-7: Applied Mathematics 1413 Chapter 7.1-: Applied Mathematics 1413 Chapter 7.1: Applied Mathematics 1413 Chapter 7.: 7.17.2 Volumes by Slicing

981 14.2 Double Integrals and Volume EXAMPLE 5 Find the volume of the solid region boanded above by the paraboloid and below by the plane Volume of a Region Bounded by Two Surfaces Mane as shown in Figure 14.20. Paraboloida 0, Figure 14.20 Solution Equating z-values, you can determine that the intersection of the two surfaces occurs on the right circular cylinder given by So, the region R in the xy-plane is a cirele, as shown in Figure 14.21. Because the volume of the solid region is the difference between the volume under the paraboloid and the volume under the plane, you have Volume (volume under paraboloid)- (volume under plane) (1 - y) d dy dy 上0Sysl dy figaure 14.21 // -GAS), [1.. (2y-1)リ1/2dy de " (I)鬧 Wallis's Formala 32

Show that using the substitution x = 2 tan theta Find the area enclosed by the curves y1(x) = 5x - x2 and y2 (x) = x. Now find the volume when the region above is rotated around the x-axis. Find the volume of the solid whose base is the circle of radius 2 centered at (0, 0) in the x-y plane and whose cross section perpendicular to the x-axis is a right triangle with hypotenuse in the x-y plane. Recall the exponential density function is f(x) = It the mean for this distribution is 4, find P(4