Lê Văn Chánh - k lvchanh@hcmus.edu.vn University of Science Contents 1 Topic 2. Double Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Topic 3. Line Integrals Type 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Topic 4. Line Integrals Type 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Topic 5. Differential Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 ic si s Topic 1. Multivariate Functions & Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . { Topic 1. Multivariate Functions & Partial Derivatives at (a) Draw level curves he m (b) Unconstrained/ constrained optimazation problems (c) Directional derivative (a) Find the volume of the solid inside the sphere x2 + y 2 + z 2 = 16 and outside the cylinder ha x2 + y 2 = 4. w (b) Find the volume of the solid bounded by the paraboloid z = 1 + 2x2 + 2y 2 and the plane z = 7 in the p x2 + y 2 and below the sphere x2 + y 2 + z 2 = 1. de re (c) Above the cone z = d first octant. (d) Find the volume of the solid bounded by the paraboloids z = 6 − x2 − y 2 and z = 2x2 + 2y 2 . w on (e) Find the volume of the solid that is enclosed by the cone z = p x2 + y 2 and the sphere x2 + y 2 + z 2 = 2. (f) Find the volume of the solid that lies between the paraboloid z = x2 +y 2 and the sphere x2 +y 2 +z 2 = 2. Li A kê (g) Find the volume of the region E that lies between the paraboloid z = 24 − x2 − y 2 and the cone p z = 2 x2 + y 2 . { Topic 3. Line Integrals (Type 1) 1. The base of a circular fence with radius 10m is given by x = 10 cos t, y = 10 sin t. The height of the fence at position (x, y) is given by the function h(x, y) = 4 + 0.01 x2 − y 2 , so the height varies Exercise 2. from 3m to 5m. Suppose that 1L of paint covers 100m2 . Sketch the fence and determine how much paint you will need if you paint both sides of the fence. 2. Tom Sawyer is whitewashing a picket fence. The bases of the the fenceposts are arranged in the xy -plane as the quarter circle x2 + y 2 = 25, x, y ≥ 0, and the height of the fencepost at point (x, y) is given Calculus III Page 1/2 LATEX by LE VAN CHANH tm Exercise 1. at { Topic 2. Double Integrals Lê Văn Chánh - k lvchanh@hcmus.edu.vn University of Science by h(x, y) := 10 − x − y (units are feet). (i) Give a rough sketch of the fence in R3 . (ii) Use a scalar line integral to find the area of one side of the fence. 3. The base of the fence runs along a quarter-circle of radius 2. The height of the fence is given by the s function f (x, y) = 3 − x − y. Using a scalar line integral to find the area of one side of the fence. (a) Find the work done by the force field F(x, y) = x2 y, xy 2 in moving an object along an at Exercise 3. ic si { Topic 4. Line Integrals (Type 2) he m circle x2 + y 2 = 1. (b) Find the work done by the force field F(x, y) = x2 i + yex j on a particle that moves once around the at unit circle centered at the origin (oriented clockwise). tm ha Exercise 4. Solve the differential equation. √ 2 (b) 2yey y 0 = 2x + 3 x. (c) y 0 + xey = 0. w on de re d w (a) (ey − 1) y 0 = 2 + cos x. Li A kê LATEX by LE VAN CHANH { Topic 5. Differential Equation Calculus III Page 2/2