A water tank in the shape of a hemispherical bowl of radius 5 m is filled with water to a depth of 3 m. How much work is required to pump all the water over the top of the tank? (The density of water is 1000 kg/m3). (Use symbolic notation and fractions where needed.) W =
Normal vectors Definition. A vector n→ is called normal to a surface (S) at a point M on (S) if n→ is orthogonal to the tangent plane of (S) at M. A unit normal vector is a normal vector of length 1. Problem. Find the unit normal vector n→ to the surface z = 2.2 − x2 − 1.1y2 at the point M(0.6, 0.4, 1.664), given that n→ is pointing upward (namely n→, Oz→ < 90∘).
Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis. y = 1/x, y = 0, x = 6, x = 11 23 132π 17 66π 17 132π 5 66π 5 132π
Use the shell method to write and evaluate the definite integral that represents the volume of the solid generated by revolving the plane region about the x-axis. y = 1/x
Find the volume of a solid of revolution formed by revolving the region bounded above by f(x) = x + 1 and below by the x-axis over the interval [0, 2] around the line y = −3. Answer exactly. units3
Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line y = 7. y = x y = 5 x = 0
Use the shell method to write and evaluate the definite integral that represents the volume of the solid generated by revolving the plane region about the y-axis. y = 2x − 9, y = 0, x = 5
Find the volume of the solid generated by revolving the shaded region about the x-axis. The volume of the solid is cubic units. (Type an exact answer, using π as needed.)
Let R be the region bounded by the graph of the function f(x) = x2 and the curves y = 0 and x = 2. Let S be the three-dimensional solid obtained by revolving the region R around the x-axis. In this problem, we use the Washer Method to approximate the volume of the solid S using a Riemann Sum.
Find the volume of the solid obtained by rotating the region bound by y = x2 16 + 4, y = 2 − x2 16 between x = 0 and x = 4 about the y-axis. Suppose we revolve the graphs of y = 4 − x2 and y = 5 for 0 ≤ x ≤ 2. Find the volume if the rotation occurs (a) about the line y = 5. (b) about the x-axis.
Consider a region R bounded by the functions f(x) = x2 + 1 and g(x) = x over the interval [0, 2]. (a) Draw an accurate graph of the region R. (b) Find the area of the region. (c) A solid of revolution is obtained by revolving a plane region bounded by a curve f(x), x ∈ [a, b], about x-axis. The volume of such solid is as V = π∫a b [f(x)]2 dx. Calculate the volume of the solid generated by the region R revolving about the x-axis.