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Mathematics LibreTexts

7.6E: Exercises

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    25945
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    Exercise \(\PageIndex{1}\): Mass

    In the following exercises, the region \(R\) occupied by a lamina is shown in a graph. Find the mass of \(R\) with the density function \(\rho\).

    1. \(R\) is the triangular region with vertices \((0,0), \space (0,3)\), and \((6,0); \space \rho (x,y) = xy\).

    A right triangle bounded by the x and y axes and the line y = negative x/2 + 3.

    Answer

    \(\frac{27}{2}\)

    3. \(R\) is the triangular region with vertices \((0,0), \space (1,1)\), and \((0,5); \space \rho (x,y) = x + y\).

    A triangle bounded by the y axis, the line x = y, and the line y = negative 4x + 5.

    4. \(R\) is the rectangular region with vertices \((0,0), \space (0,3), \space (6,3) \) and \((6,0); \space \rho (x,y) = \sqrt{xy}\).

    A rectangle bounded by the x and y axes and the lines x = 6 and y = 3.

    Answer

    \(24\sqrt{2}\)

    5. \(R\) is the rectangular region with vertices \((0,1), \space (0,3), \space (3,3)\) and \( (3,1); \space \rho (x,y) = x^2y\).

    A rectangle bounded by the y axis, the lines y = 1 and 3, and the line x = 3.

    6. \(R\) is the trapezoidal region determined by the lines \(y = - \frac{1}{4}x + \frac{5}{2}, \space y = 0, \space y = 2\), and \(x = 0; \space \rho (x,y) = 3xy\).

    A trapezoid bounded by the x and y axes, the line y = 2, and the line y = negative x/4 + 2.5.

    Answer

    \(76\)

    7. \(R\) is the trapezoidal region determined by the lines \(y = 0, \space y = 1, \space y = x\) and \(y = -x + 3; \space \rho (x,y) = 2x + y\).

    A trapezoid bounded by the x axis, the line y = 1, the line y = x, and the line y = negative x + 3.

    8. \(R\) is the disk of radius \(2\) centered at \((1,2); \space \rho(x,y) = x^2 + y^2 - 2x - 4y + 5\).