Sect 15.5 Enhanced Assignment

Instructor: Shijun Zheng Course: Spring 2023 - Calculus III (MATH2243-A-B) Student: Jalen Wilson Date: 07/11/24 1. Assignment: Sect 15.5 Enhanced Assign Find the volume of the tetrahedron shown using the order dz dx dy. The tetrahedron is bounded by the planes z = ​0, y = 16​, x = ​0, and z = y − x. 16 Complete the triple integral below used to find the volume of the tetrahedron. Note the order of integration dz dx dy. 16 y y−x ∫ ∫ ∫ 0 0 0 V= 2048 The volume of the tetrahedron is 3 dz dx dy 3 unit(s) . ​(Type an integer or a simplified​fraction.) 2 4 2 2. Evaluate the integral ∫∫∫ 2 2 2 x +y +z dz dy dx. 0 0 0 2 4 2 ∫∫∫ 2 2 2 x +y +z dz dy dx = 128 0 0 0 ​(Type a simplified fraction.) 3. Evaluate the integral. 4 π 3π ∫ ∫ ∫ y cos z dx dy dz 0 0 0 4 π 3π ∫ ∫ ∫ y cos z dx dy dz = 0 0 0 3π 3 2 sin (4) ​(Type an exact​answer.) 4. Evaluate the iterated integral. 7 2x x − 2 −3 0 ∫ ∫ ∫ 7 2x x − 2 −3 0 ∫ ∫ ∫ y dz dy dx y dz dy dx = − 80 ​(Simplify your​answer.) 5. 4 16 16 − y ∫ ∫ ∫ Here is the region of integration of the integral − 4 x2 a. ∫ ∫ ∫ − 4 x2 dz dy dx = 0 b. ∫ ∫ ∫ − 4 x2 dz dy dx = 0 ∫ ∫ ∫ − 4 x2 dz dy dx = 0 4 16 16 − y d. ∫ ∫ ∫ − 4 x2 dz dy dx = 0 4 16 16 − y e. ∫ ∫ ∫ − 4 x2 0 4 dz dy dx = 2 16 − x 16 − z ∫ ∫ ∫ −4 0 x 16 − z ∫ − ∫ 16 − z 16 16 − z ∫ ∫ 0 0 16 16 − y ∫ ∫ 0 0 16 y ∫ dy dx dz 2 x y ∫ − dx dy dz y y ∫ − ∫ − dy dz dx 2 16 − z ∫ 0 16 (4, 16, 0) 4 0 4 16 16 − y c. ( − 4, 16, 0) 16 4 16 16 − y 16 −4 equivalent integral in the following orders. a. dy dz dx b. dy dx dz c. dx dy dz d. dx dz dy e. dz dx dy 4 16 16 − y 16 dz dy dx. Rewrite the integral as an 0 16 − y ∫ y dx dz dy y 0 dz dx dy