Multivariable calculus double integration volume questionUse a double integral to find the volume of the solid bounded by graphs of the equations given by: z = x y 2 , where: z &gt; 0 x &gt; 0 5 x &lt; y &lt; 2 My problem is finding the limits of integration. I know my f ( x , y )... my guess is my y integration is going from 5x going to 2. But how do I find the x limitations?

Question

Multivariable calculus double integration volume questionUse a double integral to find the volume of the solid bounded by graphs of the equations given by:                    z        =        x                  y          2                ,         where:                     z        &amp;gt;        0                                  x        &amp;gt;        0                                  5        x        &amp;lt;        y        &amp;lt;        2            My problem is finding the limits of integration. I know my   f  (  x  ,  y  )... my guess is my y integration is going from 5x going to 2. But how do I find the x limitations?

Skylar Beard · Accepted Answer

Step 1  0  &amp;lt;  5  x  &amp;lt;  y  &amp;lt;  2  ,  0  &amp;lt;  z  &amp;lt;  x      y    2      ⟺    x  ∈  (  0  ,      2    5    )  ,  y  ∈  (  5  x  ,  2  )  ,  z  ∈  (  0  ,  x      y    2    )Step 2The lower bound on x is 0, the upper bound is y/5, which in turn has an upper bound of 2/5.                    ∴                  ∭                      0            &amp;lt;            5            x            &amp;lt;            y            &amp;lt;            2            ,            0            &amp;lt;            z            &amp;lt;            x                          y              2                                      d        ⁡        z        d        ⁡        y        d        ⁡        x                            =                  ∫          0                      2                          /                        5                                                    ∫                      5            x                    2                x                  y          2                d        ⁡        y        d        ⁡        x                                          =                  ∫          0          2                          ∫          0                      y                          /                        5                          x                  y          2                d        ⁡        x        d        ⁡        y

Use a double integral to find the volume of the solid bounded by graphs of the equations given by: z=xy^2, where: z>0, x>0, 5x<y<2.

Open question

Answer & Explanation

New Questions in High school geometry