My teacher gave us a review for our final and I have the answers to all of the questions but two which i am having a little difficulty working out. Please help me with these problems by showing how they work out, not just the answer. Thanks so much.
1. Find the volume of the solid under the graph of F(x,y) over the region R. F(x,y) = (x-y)^2; R is the region bounded by the graphs of y=x, y=2, and x=0.
2. The life expectancy (in years) of a certain type of computer chip is a continuous random variable with probability density function:
F(x) = 1/5e^(x/5) if x>0 and o otherwise.
Find the probability that a randomly selected chip will last from two to five years.
1. Find the volume of the solid under the graph of F(x,y) over the region R. F(x,y) = (x-y)^2; R is the region bounded by the graphs of y=x, y=2, and x=0.
2. The life expectancy (in years) of a certain type of computer chip is a continuous random variable with probability density function:
F(x) = 1/5e^(x/5) if x>0 and o otherwise.
Find the probability that a randomly selected chip will last from two to five years.
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Your region is a triangle in the xy-plane with vertices at (0,0), (0,2), and (2,2). You can use as limits
0 ≤ y ≤ 2, 0 ≤ x ≤ y in the order dxdy or
0 ≤ x ≤ 2, x ≤ y ≤ 2 in the order dydx.
Neither is "simpler" than the other, so I guess I'd use the first ones. The volume is
2 y
∫ .∫ (x² - 2xy + y²) dx dy = 4/3
00
I'm assuming the actual integration is easy.
2.) By definition, P(2 ≤ X ≤ 5) =
5
∫ F(x) dx
2
So the probability is
5
∫ (1/5) e^(x/5) dx = e^(x/5) {evaluate at 5 and 2} = e - e^(2/5).
2
0 ≤ y ≤ 2, 0 ≤ x ≤ y in the order dxdy or
0 ≤ x ≤ 2, x ≤ y ≤ 2 in the order dydx.
Neither is "simpler" than the other, so I guess I'd use the first ones. The volume is
2 y
∫ .∫ (x² - 2xy + y²) dx dy = 4/3
00
I'm assuming the actual integration is easy.
2.) By definition, P(2 ≤ X ≤ 5) =
5
∫ F(x) dx
2
So the probability is
5
∫ (1/5) e^(x/5) dx = e^(x/5) {evaluate at 5 and 2} = e - e^(2/5).
2