Net area from graphs The accompanying figure shows four regions bounded by the graph of y = x sin x: R 1 , R 2 , R 3 , and R 4 , whose areas are 1, π − 1 , π + 1, and 2 π − 1, respectively. (We verify these results later in the text.) Use this information to evaluate the following integrals. 37. ∫ 0 π x sin x d x
Net area from graphs The accompanying figure shows four regions bounded by the graph of y = x sin x: R 1 , R 2 , R 3 , and R 4 , whose areas are 1, π − 1 , π + 1, and 2 π − 1, respectively. (We verify these results later in the text.) Use this information to evaluate the following integrals. 37. ∫ 0 π x sin x d x
Solution Summary: The author calculates the net area of the region bounded by the graph of f and the x -axis using the given figure.
Net area from graphsThe accompanying figure shows four regions bounded by the graph of y = x sin x: R1, R2, R3, and R4, whose areas are 1, π − 1, π + 1, and 2π − 1, respectively. (We verify these results later in the text.) Use this information to evaluate the following integrals.
The OU process studied in the previous problem is a common model for interest rates.
Another common model is the CIR model, which solves the SDE:
dX₁ = (a = X₁) dt + σ √X+dWt,
-
under the condition Xoxo. We cannot solve this SDE explicitly.
=
(a) Use the Brownian trajectory simulated in part (a) of Problem 1, and the Euler
scheme to simulate a trajectory of the CIR process. On a graph, represent both the
trajectory of the OU process and the trajectory of the CIR process for the same
Brownian path.
(b) Repeat the simulation of the CIR process above M times (M large), for a large
value of T, and use the result to estimate the long-term expectation and variance
of the CIR process. How do they compare to the ones of the OU process?
Numerical application: T = 10, N = 500, a = 0.04, x0 = 0.05, σ = 0.01, M = 1000.
1
(c) If you use larger values than above for the parameters, such as the ones in Problem
1, you may encounter errors when implementing the Euler scheme for CIR. Explain
why.
#8 (a) Find the equation of the tangent line to y = √x+3 at x=6
(b) Find the differential dy at y = √x +3 and evaluate it for x=6 and dx = 0.3
Q.2 Q.4 Determine ffx dA where R is upper half of the circle shown below.
x²+y2=1
(1,0)
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Numerical Integration Introduction l Trapezoidal Rule Simpson's 1/3 Rule l Simpson's 3/8 l GATE 2021; Author: GATE Lectures by Dishank;https://www.youtube.com/watch?v=zadUB3NwFtQ;License: Standard YouTube License, CC-BY