7(x + y) ex2 − y2 dA, R where R is the rectangle enclosed by the lines x − y = 0, x − y = 7, x + y = 0, and x + y = 6

Question

7(x + y) ex2 − y2 dA, R where R is the rectangle enclosed by the lines x − y = 0, x − y = 7, x + y = 0, and x + y = 6

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Diễm Thu 2 months 2021-08-31T14:01:11+00:00 1 Answers 0 views 0

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    2021-08-31T14:02:47+00:00

    Answer:

    \int\limits {\int\limits_R {7(x + y)e^{x^2 - y^2}} \, dA = \frac{1}{2}e^{42}  -\frac{43}{2}

    Step-by-step explanation:

    Given

    x - y = 0

    x - y = 7

    x + y = 0

    x + y = 6

    Required

    Evaluate \int\limits {\int\limits {7(x + y)e^{x^2 - y^2}} \, dA

    Let:

    u=x+y

    v =x - y

    Add both equations

    2x = u + v

    x = \frac{u+v}{2}

    Subtract both equations

    2y = u-v

    y = \frac{u-v}{2}

    So:

    x = \frac{u+v}{2}

    y = \frac{u-v}{2}

    R is defined by the following boundaries:

    0 \le u \le 6  ,  0 \le v \le 7

    u=x+y

    \frac{du}{dx} = 1

    \frac{du}{dy} = 1

    v =x - y

    \frac{dv}{dx} = 1

    \frac{dv}{dy} = -1

    So, we can not set up Jacobian

    j(x,y) =\left[\begin{array}{cc}{\frac{du}{dx}}&{\frac{du}{dy}}\\{\frac{dv}{dx}}&{\frac{dv}{dy}}\end{array}\right]

    This gives:

    j(x,y) =\left[\begin{array}{cc}{1&1\\1&-1\end{array}\right]

    Calculate the determinant

    det\ j = 1 * -1 - 1 * -1

    det\ j = -1-1

    det\ j = -2

    Now the integral can be evaluated:

    \int\limits {\int\limits {7(x + y)e^{x^2 - y^2}} \, dA becomes:

    \int\limits {\int\limits {7(x + y)e^{x^2 - y^2}} \, dA = \int\limits^6_0 {\int\limits^7_0 {7ue^{x^2 - y^2}} \, *\frac{1}{|det\ j|} * dv\ du

    x^2 - y^2 = (x + y)(x-y)

    x^2 - y^2 = uv

    So:

    \int\limits {\int\limits {7(x + y)e^{x^2 - y^2}} \, dA = \int\limits^6_0 {\int\limits^7_0 {7ue^{uv}} *\frac{1}{|det\ j|}\, dv\ du

    \int\limits {\int\limits {7(x + y)e^{x^2 - y^2}} \, dA = \int\limits^6_0 {\int\limits^7_0 {7ue^{uv}} *|\frac{1}{-2}|\, dv\ du

    \int\limits {\int\limits {7(x + y)e^{x^2 - y^2}} \, dA = \int\limits^6_0 {\int\limits^7_0 {7ue^{uv}} *\frac{1}{2}\, dv\ du

    Remove constants

    \int\limits {\int\limits {7(x + y)e^{x^2 - y^2}} \, dA = \frac{7}{2}\int\limits^6_0 {\int\limits^7_0 {ue^{uv}} \, dv\ du

    Integrate v

    \int\limits {\int\limits {7(x + y)e^{x^2 - y^2}} \, dA = \frac{7}{2}\int\limits^6_0  \frac{1}{u} * {ue^{uv}} |\limits^7_0  du

    \int\limits {\int\limits {7(x + y)e^{x^2 - y^2}} \, dA = \frac{7}{2}\int\limits^6_0  e^{uv} |\limits^7_0  du

    \int\limits {\int\limits {7(x + y)e^{x^2 - y^2}} \, dA = \frac{7}{2}\int\limits^6_0  [e^{u*7} -   e^{u*0}]du

    \int\limits {\int\limits {7(x + y)e^{x^2 - y^2}} \, dA = \frac{7}{2}\int\limits^6_0  [e^{7u} -   1]du

    Integrate u

    \int\limits {\int\limits {7(x + y)e^{x^2 - y^2}} \, dA = \frac{7}{2} * [\frac{1}{7}e^{7u} -   u]|\limits^6_0

    Expand

    \int\limits {\int\limits {7(x + y)e^{x^2 - y^2}} \, dA = \frac{7}{2} * ([\frac{1}{7}e^{7*6}  - 6) -(\frac{1}{7}e^{7*0} -  0)]

    \int\limits {\int\limits {7(x + y)e^{x^2 - y^2}} \, dA = \frac{7}{2} * ([\frac{1}{7}e^{7*6}  - 6) -\frac{1}{7}]

    Open bracket

    \int\limits {\int\limits {7(x + y)e^{x^2 - y^2}} \, dA = \frac{7}{2} * [\frac{1}{7}e^{7*6}  - 6 -\frac{1}{7}]

    \int\limits {\int\limits {7(x + y)e^{x^2 - y^2}} \, dA = \frac{7}{2} * [\frac{1}{7}e^{7*6}  -\frac{43}{7}]

    \int\limits {\int\limits {7(x + y)e^{x^2 - y^2}} \, dA = \frac{7}{2} * [\frac{1}{7}e^{42}  -\frac{43}{7}]

    Expand

    \int\limits {\int\limits {7(x + y)e^{x^2 - y^2}} \, dA = \frac{1}{2}e^{42}  -\frac{43}{2}

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