Evaluate the Iterated Integral

Section Exercise 1. Evaluate: R 2 R 2x Exercise 2. Evaluate: R 3 R y2 0 1 x2 π/6 (x3 + 4y) dy dx. (Swok Sec 17.1 Ex 3) 2y cos x dx dy. (Swok Sec 17.1 Ex 4) Class Exercise 1. Evaluate the iterated integral. (#2,4,6) R1 R2 R 2 R 2y R 1 R ev √ (a) 0 2x (x − y) dy dx (b) 0 y xy dx dy (c) 0 0 1 + ev dw dv Definition: A plane region D is said to be of type I if it lies between the graphs of two consecutive functions of x, that is, D = { (x, y) | a ≤ x ≤ b, g1 (x) ≤ y ≤ g2 (x) } Theorem: If f is continuous on a type I region D such that then RR D D = { (x, y) | a ≤ x ≤ b, g1 (x) ≤ y ≤ g2 (x) }, R b R g2 (x) f (x, y) dA = a g1 (x) f (x, y) dy dx. Definition: A plane region D is said to be of type II if it can be expressed as D = { (x, y) | c ≤ y ≤ d, h1 (y) ≤ x ≤ h2 (y) }, where h1 and h2 are continuous. Formula: RR f (x, y) dA = D R d R h2 (y) c h1 (y) f (x, y) dx dy, where D is a type II region. R4 R2 Exercise 3. Given 0 √y y cos x5 dx dy, reverse the order of integration and evaluate the resulting integral. (Swok Sec 17.1 Ex 7) Exercise 4. Calculate RR R (sin x)/x dA, where R is the triangle in the xy-plane bounded by the x-axis, the line y = x, and the line x = 1. (Hass Sec 15.2 Ex 2) Class 2. Evaluate the double integral. (#8,10) RR Exercise y 2 (a) x5 +1 dA, D = { (x, y) | 0 ≤ x ≤ 1, 0 ≤ y ≤ x } D RR 3 (b) x dA, D = { (x, y) | 1 ≤ x ≤ e, 0 ≤ y ≤ ln x } D ExerciseRR 5. Let R be the region in the xy-plane bounded by the graphs of y = x2 and y = 2x. Evaluate R (x3 + 4y) dA. (Swok Sec 17.1 Ex 5) √ √ Exercise 6. Let R be the region bounded by the graphs of the equations y = x, y = 3x − 18, and RR y = 0. If f is an arbitrary continuous function on R, express the double integral f (x, y) dA in terms of iterated integrals. (Swok Sec 17.1 Ex 6) R Class RR (a) D RR (b) D RR (c) D Exercise 3. Evaluate the double integral. (#18,20,22) (x2 + 2y) dA, D is bounded by y = x, y = x3 , x ≥ 0 p xy 2 dA, D is enclosed by x = 0 and x = 1 − y 2 2xy dA, D is the triangular region with vertices (0,0), (1,2), and (0,3). Exercise 7. Find the volume V of the solid in the first octant bounded by the coordinate planes, the paraboloid z = x2 + y 2 + 1, and the plane 2x + y = 2. (Swok Sec 17.2 Ex 3) Exercise 8. Find the volume V of the solid that lies in the first octant and is bounded by the three coordinate planes and the cylinders x2 + y 2 = 9 and y 2 + z 2 = 9. (Swok Sec 17.2 Ex 4) Class Exercise 4. Find the volume of the given solid. (#24, 26, 28, 30, 32) (a) Under the surface z = 1 + x2 y 2 and above the region x = y 2 and x = 4 (b) Enclosed by the paraboloid z = x2 + 3y 2 and the planes x = 0, y = 1, y = x, z = 0 (c) Bounded by the planes z = x, y = x, x + y = 2, and z = 0 (d) Bounded by the cylinders y 2 + z 2 = 4 and the planes x = 2y, x = 0, z = 0 in the first octant (e) Bounded by the cylinders x2 + y 2 = r2 and y 2 + z 2 = r2 Exercise 9. Sketch the region of integration for the integral R 2 R 2x (4x + 2) dy dx 0 x2 and write an equivalent integral with the order of integration reversed. (Hass Sec 15.2 Ex 3) Class Exercise 5. Sketch the region of integration and change the order of integration. (#44, 46, 48) R 1 R π/4 R2 R4 R 2 R √4−y2 f (x, y) dx dy (c) 0 arctan x f (x, y) dy dx (a) 0 x2 f (x, y) dy dx (b) −2 0 Class√ Exercise 6. Evaluate the integral by reversing the order of integration. (#50, 52, 54) R 8 R 2 x4 R1 R1 R π R √π dx dy cos(x2 ) dx dy (b) 0 x ex/y dy dx (c) 0 √ (a) 0 3 y e y Homework: 1, 5, 13, 17, 19, 25, 31-47 (every 4th), 55, 59, 61, 65