Double integrals—transformation given To evaluate the following integrals, carry out these steps. a. Sketch the original region of integration R in the xy-plane and the new region S in the uv-plane using the given change of variables. b. Find the limits of integration for the new integral with respect to u and v. c. Compute the Jacobian. d. Change variables and evaluate the new integral. 29. ∬ R x 2 x + 2 y d A , where R = {( x , y ): 0 ≤ x ≤ 2, – x /2 ≤ y ≤ 1 – x }; use x = 2 u , y = v – u.
Double integrals—transformation given To evaluate the following integrals, carry out these steps. a. Sketch the original region of integration R in the xy-plane and the new region S in the uv-plane using the given change of variables. b. Find the limits of integration for the new integral with respect to u and v. c. Compute the Jacobian. d. Change variables and evaluate the new integral. 29. ∬ R x 2 x + 2 y d A , where R = {( x , y ): 0 ≤ x ≤ 2, – x /2 ≤ y ≤ 1 – x }; use x = 2 u , y = v – u.
Double integrals—transformation givenTo evaluate the following integrals, carry out these steps.
a. Sketch the original region of integration R in the xy-plane and the new region S in the uv-plane using the given change of variables.
b. Find the limits of integration for the new integral with respect to u and v.
c. Compute the Jacobian.
d. Change variables and evaluate the new integral.
29.
∬
R
x
2
x
+
2
y
d
A
, where R = {(x, y): 0 ≤ x ≤ 2, –x/2 ≤ y ≤ 1 – x}; use x = 2u, y = v – u.
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
Use the given transformation to evaluate the integral.
10xy dA, where R is the region in the first quadrant bounded by the lines y = =x and y = x
and the hyperbolas xy = - and xy = ;
x = u/v, y = v
Find the area of the region in the first quadrant bounded by the curves y
U
y = x, and y = 4x using the change of variables x =
integration before and after the transformation.
V
1
7
X
Y
X
y = uv. Sketch the region of
2
Use the given transformation to evaluate the integral.
16. (8x + 8y) dA, where R is the parallelogram with vertices (-2, 6), (2, -6), (5,-3), and (1, 9);
X =
= ¹/(u + v), y = ¹/(v - 3u)
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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