Section 7.6. Consider a planar lamina of uniform density ρ, bounded by the graphs of y = f (x) and y = g(x), a x b (with f (x) g(x) on the interval).

Transcription

1 February 6, 2018 In these slides we will Review the theorem for the center of mass for a planar lamina. Go over an example calculating the center of mass. Present the Theorem of Pappus.

2 Consider a planar lamina of uniform density ρ, bounded by the graphs of y = f (x) and y = g(x), a x b (with f (x) g(x) on the interval).

3 Consider a planar lamina of uniform density ρ, bounded by the graphs of y = f (x) and y = g(x), a x b (with f (x) g(x) on the interval).

4 Consider a planar lamina of uniform density ρ, bounded by the graphs of y = f (x) and y = g(x), a x b (with f (x) g(x) on the interval). The center of mass (also called the centroid) of this planar lamina is ( x, ȳ) where x = M y m, ȳ = M x m. m =mass of the planar lamina M x =moment about the x-axis M y =moment about the y-axis

5 Theorem The mass of this planar lamina is m = (density)(area) = ρ b a [f (x) g(x)] dx.

6 Theorem The moment about the x-axis is ( n ) M x = lim (mass) i (distance from x-axis) i n i=1 b [ ] f (x) + g(x) = ρ (f (x) g(x)) dx 2 a

7 Theorem The moment about the x-axis is ( n ) M x = lim (mass) i (distance from x-axis) i n i=1 b [ ] f (x) + g(x) = ρ (f (x) g(x)) dx 2 a

8 Theorem The moment about the x-axis is ( n ) M x = lim (mass) i (distance from x-axis) i n i=1 b [ ] f (x) + g(x) = ρ (f (x) g(x)) dx 2 a

9 Theorem The moment about the y-axis is ( n ) M y = lim (mass) i (distance from y-axis) i n = ρ b a i=1 (f (x) g(x))x dx

10 Theorem The moment about the y-axis is ( n ) M y = lim (mass) i (distance from y-axis) i n = ρ b a i=1 (f (x) g(x))x dx

11 Theorem The moment about the y-axis is ( n ) M y = lim (mass) i (distance from y-axis) i n = ρ b a i=1 (f (x) g(x))x dx

12 Theorem Consider a planar lamina of uniform density ρ, bounded by the graphs of y = f (x) and y = g(x), a x b (f (x) g(x)) 1 The mass is given by m = ρ a b a (f (x) g(x)) dx. 2 The moment about the x-axis is b [ ] f (x) + g(x) M x = ρ (f (x) g(x)) dx. 2 3 The moment about the y-axis is M y = ρ b a (f (x) g(x))x dx. 4 The center of mass (or centroid), ( x, ȳ) is given by x = M y m, ȳ = M x m.

13 Example Find the center of mass for the planar laminar of uniform density ρ that is bounded by the graphs of y = x 2 and y = x + 6.

14 Example Find the center of mass for the planar laminar of uniform density ρ that is bounded by the graphs of y = x 2 and y = x + 6. Try this before clicking next.

15 Example Find the center of mass for the planar laminar of uniform density ρ that is bounded by the graphs of y = x 2 and y = x + 6. Try this before clicking next. First we sketch this region. To calculate the center of mass, ( x, ȳ), we need M x, M y, and m. In this example f (x) = x + 6 (top), g(x) = x 2 (bottom), a = 2, and b = 3.

16 m = ρ M x = ρ M y = ρ [(x + 6) x 2 ] dx = ρ125 6 [ ] (x + 6) + x [(x + 6) x 2 2 ] 2 [(x + 6) x 2 ]x dx = ρ dx = ρ250 3

17 m = ρ M x = ρ M y = ρ Hence we get that [(x + 6) x 2 ] dx = ρ125 6 [ ] (x + 6) + x [(x + 6) x 2 2 ] 2 [(x + 6) x 2 ]x dx = ρ dx = ρ250 3 x = M ρ125 y m = 12 ρ125 6 ȳ = M ρ250 x m = 3 ρ125 6 = ρ ρ125 = 1 2 = ρ ρ125 = 4

18 Example Find the center of mass for the planar laminar of uniform density ρ that is bounded by the graphs of y = x 2 and y = x + 6. Hence we get that the center of mass is (1/2, 4).

19 Theorem (Theorem of Pappus) Let R be a region in a plane and let L be a line in the same plane such that L does not intersect the interior of R (see figure). If r is the distance between the center of mass ( or centroid) of R and the line L, then the volume of V of the solid of revolution formed by revolving R about the line is V = 2πrA where A is the area of the region R.

20 Example Use the Theorem of Pappus to find the volume of the region bounded by y = x + 6 and y = x 2 rotated around the line x = 4.

21 Example Use the Theorem of Pappus to find the volume of the region bounded by y = x + 6 and y = x 2 rotated around the line x = 4. Note: this is the same region we calculated the center of mass for. Recall that ( x, ȳ) = (1/2, 4).

22 Example Use the Theorem of Pappus to find the volume of the region bounded by y = x + 6 and y = x 2 rotated around the line x = 4. Note: this is the same region we calculated the center of mass for. Recall that ( x, ȳ) = (1/2, 4). The Theorem of Pappus tells us that V = 2πrA where r=distance from the center of mass to the line x = 4 and A is the area of the region. Find r and A before clicking next.

23 Example Use the Theorem of Pappus to find the volume of the region bounded by y = x + 6 and y = x 2 rotated around the line x = 4.

24 Example Use the Theorem of Pappus to find the volume of the region bounded by y = x + 6 and y = x 2 rotated around the line x = 4. Here r = 4 (1/2) = 7/2.

25 Example Use the Theorem of Pappus to find the volume of the region bounded by y = x + 6 and y = x 2 rotated around the line x = 4. Here r = 4 (1/2) = 7/2. And A = 3 2 [(x + 6) x 2 ] dx =

26 Example Use the Theorem of Pappus to find the volume of the region bounded by y = x + 6 and y = x 2 rotated around the line x = 4. Here r = 4 (1/2) = 7/2. And A = 3 2 [(x + 6) x 2 ] dx = Thus, V = 2πrA = 2π( )( 6 ) = 875π 6.