Question

1. Consider the region bounded by x = y, y = 0, and y=v2 - X. Sketch the region, and find its area.

Answer 1

Given equations of the cureres aree =0 @ and y=v2- 00, y = -(x-2)3 Plotting . and ① in sy plane, we get

y = 12.7 ci azzy U The area enclosed by the curves 0,0 and are indicated in above figure.

- To determine the coordinate of B. we solve cequations O and Ⓡ . Using @ in ©, we get x=y? = (M-232 or, si=x²-qx + 9 or; 22-54+9=0 on (-1)(x-4)=0 or si = 1,9 But we can't take a=4 because @ gires :-(4-2) =-2 which is impossible

So, x=l and then y=va-1. at 26 intersects B (1,1) Hence curves Now, the area of indicated region OBACO. = area of the region OBCO + . area of the region EBAC

- Siyax [where x=y75 +5, šau [whare yo vza] Satax +j Vz-x de +]+\-(2-23 d. -x ľa (2-1)= - t À ty? -(2-x) -> -d(2-x) = d.x]

کے دو . ها ها ها ها هم د | مرة ت ا ا أسرع ما تام ) . ترافر - 1 1 - 1 (1-9 [-] 1 کے M1 +۱ -2