Question

Show all of your work. Label the axes and scale on your graph. The sketch should include at least three . to five points including the points of intersection and at least one point in between The integration techniques used should be clear and easy to follow step by step with all techniques shown in detail, including the substitution method. . . All steps must be shown in your own writing to receive any credit. Circle or box your final answers. Draw a line separating each problem. . The region R is bounded by the x-axis and y = V16 - x 2. a) (0.75 point) Sketch the bounded region R. Label your graph. b) (0.75 point) Set up the iterated integral to solve for the area of the bounded region using either the Rx region or RY region. Do not integrate. c) (1.25 point) Evaluate the integral using polar coordinates for the region R. sec (x 2 + y 2) tan(x 2 + y 2) da R

Answer 1

R y d=9 YE116-1² 8 SS See (x² + y 2) tan (x² + y 2) dA. Sams) a) Region IR : Region is x-anis & y = (16-x2 6 -5 -4 -3 Shaded portion is the region of Integration which is the upper half of the circle with radius 4. oxyz √16-212 -4 << 4. up the Integral in x-y co-ordinates, S Sec (x² + y 2) tan (x2 + y 2) dA R to "2 2.3 4 5 6 u b) Set 4 See 6x² + y2) dan (x² + y²) dy dx x=-4 y=0

co-ordinates - r Sino rde do O Evaluate the Integral in Polar put a = r coso y= x² + y² = ez & dxdy Region of Integration polar to-ordinates is i oseca orost G SS See (e tan (4²) e de do o-o e o in ㅠ 고 4 S S See (42) tan (82) Ede/do. &=0 II {avaluate It Je = See (e) tanc&?) e de 4 put e²=t ar dr = dt > Ede Ede=at Ip - I See its tan its dt. २ Seelts fanctdt. Seclts ㅗ 2. Rebsubstitute value of t=ą2.

Seel &2) Apply the limits from I [See (42) a E =0 to 4 See (16) Secco Substitute In in Eqna we Get ㅠ s (See (16) - 8410) alojao هجي 2 See (16) - Lecco)] [o] See (10-Seelo) ] T = π Sec (16) - Lecco) or/ I [_-2.0492125] - 3.2110415