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Triple Integral | IIT JAM 2016 | Question 15

This is a beautiful problem from IIT JAM 2016, Question no.15 based on triple integral. We provide sequential hints to solve the problem.

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    Question 15 – Triple Integral (IIT JAM 2016)


    If the triple integral over the region bounded by the planes $2x+y+z=4$ $x=0$ $y=0$ $z=0$ is given by $\int\limits_0^2\int\limits_0^{\lambda(x)}\int\limits_0^{\mu(x,y)}\mathrm d z\mathrm d y\mathrm d x$ then the function $\lambda(x)-\mu(x,y)$ is

    • $x+y$
    • $x-y$
    • $x$
    • $y$

    Key Concepts


    Real Analysis

    Integral Calculus

    Triple Integral

    Check the Answer


    Answer: $\textbf{(B)} \quad y$

    IIT JAM 2016, Question No. 15

    Differential and Integral Calculus: R Courant

    Try with Hints


    Here we are given with triple integral over the region bounded by the planes $2x+y+z=4, x=0, y=0$ and $z=0$

    Now we our aim here is to find $\lambda (x) $ and $\mu(x,y)$. Now we will approach this problem by find the volume of $(x,y,z)$ based on $2x+y+z=4$ can you do this ??? (With the given information x=0, y=0, z=0)

    $2x+y+z=4$

    $\Rightarrow z=4-2x-y$

    $\Rightarrow 2x+y=4$ [as $z=0$]

    $\Rightarrow y=4-2x$

    Again,

    $2x+y+z=4$

    Now as $y=z=0$ we have $2x=4$

    Therefore $x=2$

    Now can you use this to move forward with this problem ?

    So our triple integral become,

    $\int_0^2\int_0^{(4-2x)}\int_0^{(4-2x-y)}\mathrm d z\mathrm d y\mathrm dx$

    On compairing $\lambda(x)=4-2x$ and $\mu(x,y)=4-2x-y$

    Therefore $\lambda(x)-\mu(x,y)=4-2x-4+2x+y=y$ (ANS)

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