Solução_Calculo_Stewart_6e

F.
376 ¤ CHAPTER 8 FURTHER APPLICATIONS OF INTEGRATION
TX.10
15. The centroid of this circle, (1, 0), travels a distance 2π(1) when the lamina is rotated about the y-axis. The area of the circle
is π(1) 2 . So by the Theorem of Pappus, V = A(2πx) =π(1) 2 2π(1) = 2π 2 .
17. x =100 ⇒ P = 2000 − 0.1(100) − 0.01(100) 2 = 1890
19. f(x) =
Consumer surplus = 100
[p(x) − P ] dx = 100 0 0 2000 − 0.1x − 0.01x 2 − 1890 dx
π
20 sin π
10 x if 0 ≤ x ≤ 10
0 if x10
(a) f(x) ≥ 0 for all real numbers x and
∞
−∞ f(x) dx = 10
0
= 110x − 0.05x 2 − 0.01 x3 100
=11,000 − 500 − 10,000 ≈ $7166.67
3 0
3
π
sin π
x dx = π · 10
20 10 20 π − cos π x 10
= 1 (− cos π +cos0)= 1 (1 + 1) = 1
10 0 2 2
Therefore, f is a probability density function.
(b) P (X