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- Suppose that r=12 cm and h=15 cm in the right circular cylinder. Find the exact and approximate a lateral area. b total area. c volume.Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve y=ex and the line x=ln14 about the line x=ln14.A pulse (gas) travelling along a fine straight uniform tube filled with gas causes the density at any time t and distance x from the origin where the velocity is u\index{0} to become ?\index{0}ϕ(vt-x). Prove that the velocity u (at time t and distance x from the origin) is given by
- The graphs y = sin x and y = (-x + π/2 +1) intersect only at the point (π/2 ,1). Let R be the area to the right of the y-axis (x = 0), above the curve y = sin x and below the curve y = -x + π/2 + 1. What is the volume of the body of revolution as we rotate R about the y-axis?Find the volume generated by revolving the area bounded by the curve y = (1 − x2)2 and the x-axis about the y − axis. Illustrate the graph.A window has the shape of a square surmounted by a semicircle. The base of the window is measured as having width 60 cm with a possible error in measurement of 0.1 cm. Use differentials to estimate the maximum error possible in computing the area of the window.
- A water container with a square base, whose cross section is 4ft2 and whose height is 9 ft is fully filled with an unknown fluid. It has a hole at the base of radius 1 inch. When will the container be empty? Use application of First Order ODE Orifice.If x ≥ 0, find the volume of the solid obtained by rotating th reign enclose by the graphs about the line y=-8.Find the volume generated by revolving about the line x – 4 = 0 the area bounded by the curve x2 = 8y and the line y – 2 = 0. Show full complete solution
- A hole of radius 3 is bored through the center of a sphere of radius 5. Let the cross-section of the sphere through the center be the circle x^2+ y^2 = 25 cross-sections of the hole be defined by lines x =− 3 and x =3. Set up the definite integral that will solve for the volume of the remaining portion of the sphere using: Vertical strips? Horizontal strips?Each integral represents the volume of a solid.Describe the solid. ∫0π/42 π (π - x) (Cos x - Sinx) dxA mass of 1 kgkg is attached to the end of a spring whose restoring force is 180 NmNm. The mass is in a medium that exerts a viscous resistance of 156 NN when the mass has a velocity of 6 msms. The viscous resistance is proportional to the speed of the object. Suppose the spring is stretched 0.07 mm beyond the its natural position and released. Let positive displacements indicate a stretched spring, and suppose that external vibrations act on the mass with a force of 7sin(3t)7sin(3t) NN at time tt seconds. Find an function to express the steady-state component of the object's displacement from the spring's natural position, in mm after tt seconds. (Note: This spring-mass system is not "hanging", so there is no gravitational force included in the model.)