GATE 2014 EE (EE2) – Question 36

Engineering Mathematics · Calculus: Mean value theorems, Theorems of integral calculus, Evaluation of definite and improper integrals, Partial derivatives, Multiple integrals, Fourier series · 2 marks · Multiple choice

For $\int_0^8\int_{y/2}^{y/2+1}(2x-y)/2\,dx\,dy$, substitute u = (2x−y)/2 and v = y/2. Find the transformed integral.

$\int_0^4\int_0^2 2u\,du\,dv$
$\int_0^4\int_0^1 2u\,du\,dv$
$\int_0^4\int_0^1 u\,du\,dv$
$\int_0^4\int_0^2 u\,du\,dv$

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Correct answer: (B) $\int_0^4\int_0^1 2u\,du\,dv$

Explanation

The substitution gives x=u+v,y=2v. The original bounds0≤y≤8 and y/2≤x≤y/2+1become0≤v≤4and0≤u≤1. The Jacobian determinant |∂(x,y)/∂(u,v)| is2. Since the integrand is u,the transformed integral is ∫0^4∫0^1 2u du dv, **B**.