1.Find the value if the iterated integral: a) \\(\\int_{0}^{2}\\int_{2y}^{4}sin(

Question

1.Find the value if the iterated integral:

a)       (int_{0}^{2}int_{2y}^{4}sin(x^{2}) dxdy)     

b)       (int_{0}^{4}int_{y/2}^{2}e^{x^{2}} dxdy)     

c)       (int_{0}^{1}int_{arctan y}^{pi/4}1/(cos^{10}x) dxdy)     

2.  Consider the transformation T: R2 ---- R2, T(u,v) = (x,y) = (u - 2v, u + v) and the region R on the plane xy bounded by the triangle with vertexs A = (0,0), B= (1,1), C = (-2,1).

  a) Find the region S on the plane uv so that T(S) = R.

  b) Evaluate the double integral of x+2y using the change of variable theorem.  

Explanation / Answer

1a) ANSWER IS 0.4894

PROCESS IS

Let I = int. sin(x^2) dx

let, x^2 = z
=> 2x dx = dz
=> dx = dz/2x
=> dx = dz/2sqrt(z)

so, I = int. sin(x^2) dx
= int. sin z . dz/2.(z)^1/2
= 1/2 int . sin z . z^-1/2 dz

FROM HERE EVERY TERM WILL BE IN TERMS OF y

1B)

ANSWER IS 53.59

1c) ANSWER IS 0.4852

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