Determine the centroid of the region bounded by the graphs

Assignment Help Mathematics

Reference no: EM132345262

Assignment -

Part A -

Q1. Evaluate the integrals:

(a) ₀∫^½ x²/√(1-x²) dx

(b) ∫dx/(x²+1)^3/2

Q2. Evaluate the integrals:

(a) ₁∫^∞ lnx/x² dx

(b) ₀∫⁹dx/√(9-x)

Q3. Compute the surface area of revolution of f(x) = x² about the x-axis over the interval [0, 4].

Q4. Determine the centroid of the region bounded by the graphs of f(x) = 2 - x, g(x) = 1/x over the interval [1, 2].

Q5. Verify that y = e^x sin2x is a solution of the second order differential equation y" - 2y' + 5y = 0.

Q6. Solve the first order linear differential equation y' + 2y = 1 + e^-x, y(0) = -4.

Q7. A lake has a carrying capacity of 1000 fish. If the fish population grows logistically with growth constant k = 0.2 week^-1, how many weeks will it take for the population to reach 500 fish if the initial population is 20 fish.

Part B -

Q1. Find the exact value of ₀∫^∞e^-4x sin xdx, and obtain a decimal approximation.

Q2. Sketch the ellipse with equation x²/a² + y²/b² = 1. Show that the area of this ellipse is πab.

Q3. Find the centroid of the region in the first quadrant bounded by the x-axis, the y-axis, the line x = 1 and the graph of y = 5e^-x. Mark the centroid on the graph.

Q4. Find the centroid of the region bounded by the y-axis, and the graphs of y = 2√x and y = 2x - 4. Mark the centroid on the graph.

Q5. Find the quadratic (second order) Maclaurin polynomial T₂(x) for f(x) = √(x+9) near a = 0. Compare the values T₂(.01) and f(.01).

Q6. Find the centroid of the region bounded by the x-axis and the graph of y = -(x-2)² + 1. Locate this centroid on the graph.

Q7. Find the general solution of the differential equation 4dy/x = - dx/y, and the particular solution if y(4) = 2.

Q8. Solve the initial value problem: (x² + 3x)y' = (2x+3)y, y(1) = 8.

Q9. The rate at which an epidemic spreads among a population of 2000 susceptible individuals follows a logistic model. If the initial number of infected individuals was 500, and after 1 week there were 855 individuals infected, determine:

(a) The equation of the number of individuals infected after t weeks.

(b) How long does it take for 1900 individuals to be infected?

Q10. A plate in the form of half an ellipse with semimajor axis = 3 and semiminor axis = 2 is submerged in a water tank, with its major axis on the surface of the tank. Determine the total hydrostatic force exerted on this plate. Consider the origin to be at the center of the ellipse on the surface of the tank, with the positive y-axis pointing downward.

Reference no: EM132345262

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