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12. The ellipses 3x 2 + 4y 2 = C all satisfy the differential equation 6x + 8y dy = 0, dx or dy 3x =− . dx 4y A family of curves that intersect these ellipses at right angles must therefore have slopes dy 4y given by = . Thus dx 3x Z Z dx dy =4 3 y x 3 ln |y| = 4 ln |x| + ln |C|. The family is given by y 3 = C x 4 . 8. y 1 3 x Fig. R-8 Let the disk have centre (and therefore centroid) at (0, 0). Its area is 9π . Let the hole have centre (and therefore centroid) at (1, 0). Its area is π . The remaining part has area 8π and centroid at (x, 0), where (9π )(0) = (8π )x + (π )(1). Thus x = −1/8. The centroid of the remaining part is 1/8 ft from the centre of the disk on the side opposite the hole. 20. y ′ + (cos x)y = 2xe− sin x , Z µ = cos x d x = sin x y(π ) = 0 d sin x (e y) = esin x (y ′ + (cos x)y) = 2x dx Z esin x y = 2x d x = x 2 + C y(π ) = 0 ⇒ 0 = π 2 + C ⇒ C = −π 2 y = (x 2 − π 2 )e− sin x . 17. 1 2 2 √ e−(x−µ) /2σ σ 2π Z ∞ 1 2 2 xe−(x−µ) /2σ d x mean = √ σ 2π −∞ f µ,σ (x) = Let z = dz = = = variance = = = x −µ σ 1 dx σ Z ∞ 1 2 (µ + σ z)e−z /2 dz √ 2π Z−∞ ∞ µ 2 e−z /2 dz = µ √ 2π −∞ E (x − µ)2 Z ∞ 1 2 2 (x − µ)2 e−(x−µ) /2σ d x √ σ 2π Z−∞ ∞ 1 2 σ 2 z 2 e−z /2 dz = σ Var(Z ) = σ √ σ 2π −∞ 9. A layer of water between depths y and y + dy has volume d V = π(a 2 − y 2 ) dy and weight d F = 9, 800π(a 2 − y 2 ) dy N. The work done to raise this water to height h m above the top of the bowl is dW = (h + y) d F = 9, 800π(h + y)(a 2 − y 2 ) dy N·m. Thus the total work done to pump all the water in the bowl to that height is Z a (ha 2 + a 2 y − hy 2 − y 3 ) dy hy 3 y 4 a a2 y 2 2 − − = 9, 800π ha y + 2 3 4 0 3 a4 2a h + = 9, 800π 3 4 8h 3 3 3a + 8h = 2450π a a + N·m. = 9, 800π a 12 3 W = 9, 800π 0 y dy Fig. 6-9 a