Question

os. Define R as the region bounded by the graph of y2x-x' and by the axis over the interval [0.21 Find the volume of the solid of revolution formed by revolving Ruround the y-axis. Hint: Use Shell Method. 16 pts) 6. Find the average value, so of the function /(x) = cos x over the interval (0, 2) and find e such that f(e) equals the average value of the function over [0, 2x) 15 pts] 7. Express the limits lim sin(x)ar as a definite integral on the interval (0.1) and then evaluate the integral.

Answer 1

a Q- Yz - 2x -k 2 Shell Methad dv 21 x dx (9,-) K2 2 카 dx = 27 xdx ((2x-x2)-0) za 12x²_x3) de -- - 2 - U V = 20 (2x?_x?) dx 3 2 za / 2x = 2a 1.23- "}-(0)) = 24.2*** 87 Aus 12 Il Q→ f(x) = cosa (0.27) Jang jo f(x). de bra 20 Casa da 27-0

fang alat [ Sine 10 ) fil- 1 sin 2a - Sino ol - 14 f(e) fang cas c c U = I, 3a 2 2 Q- A lim E sin(x; l ox nto n=1 [onl f(a) Sink. 13 [a,bl = [0,a] a=o T A ja sinx dx ) f-casa/ -Cast + Caso + L + L 2 Am