Double Integrals — Practice Problems and Worked Solutions

17 worked problems

Every step is generated and checked by a computer algebra system.

1. 1.

   Evaluate the iterated integral over the rectangle.

   $${\int }_0^1{\int }_0^{1}x+ydxdy$$

   Answer: $1$

   Step-by-step solution

   1. $$Integratewithrespecttoxfirst,theny.$$
   2. $$=1$$
2. 2.

   Evaluate the iterated integral over the rectangle.

   $${\int }_0^3{\int }_0^{2}xydxdy$$

   Answer: $9$

   Step-by-step solution

   1. $$Integratewithrespecttoxfirst,theny.$$
   2. $$=9$$
3. 3.

   Evaluate the iterated integral over the rectangle.

   $${\int }_0^2{\int }_{-1}^1{x}^2+ydxdy$$

   Answer: $\frac{16}{3}$

   Step-by-step solution

   1. $$Integratewithrespecttoxfirst,theny.$$
   2. $$=\frac{16}{3}$$
4. 4.

   Evaluate the iterated integral over the rectangle.

   $${\int }_1^3{\int }_0^{2}2x+3ydxdy$$

   Answer: $32$

   Step-by-step solution

   1. $$Integratewithrespecttoxfirst,theny.$$
   2. $$=32$$
5. 5.

   Evaluate the iterated integral over the rectangle.

   $${\int }_{-1}^1{\int }_0^{1}x{y}^{2}dxdy$$

   Answer: $\frac{1}{3}$

   Step-by-step solution

   1. $$Integratewithrespecttoxfirst,theny.$$
   2. $$=\frac{1}{3}$$
6. 6.

   Evaluate the iterated integral over the rectangle.

   $${\int }_0^2{\int }_0^2{x}^2+y^{2}dxdy$$

   Answer: $\frac{32}{3}$

   Step-by-step solution

   1. $$Integratewithrespecttoxfirst,theny.$$
   2. $$=\frac{32}{3}$$
7. 7.

   Evaluate the iterated integral over the rectangle.

   $${\int }_0^2{\int }_0^{1}x-ydxdy$$

   Answer: $-1$

   Step-by-step solution

   1. $$Integratewithrespecttoxfirst,theny.$$
   2. $$=-1$$
8. 8.

   Evaluate the iterated integral over the rectangle.

   $${\int }_0^2{\int }_0^{1}2xydxdy$$

   Answer: $2$

   Step-by-step solution

   1. $$Integratewithrespecttoxfirst,theny.$$
   2. $$=2$$
9. 9.

   Evaluate the iterated integral over the rectangle.

   $${\int }_0^1{\int }_0^2{x}^2-ydxdy$$

   Answer: $\frac{5}{3}$

   Step-by-step solution

   1. $$Integratewithrespecttoxfirst,theny.$$
   2. $$=\frac{5}{3}$$
10. 10.

    Evaluate the iterated integral over the rectangle.

    $${\int }_0^4{\int }_0^{1}3x+ydxdy$$

    Answer: $14$

    Step-by-step solution

    1. $$Integratewithrespecttoxfirst,theny.$$
    2. $$=14$$
11. 11.

    Evaluate the iterated integral over the rectangle.

    $${\int }_0^3{\int }_1^{2}xydxdy$$

    Answer: $\frac{27}{4}$

    Step-by-step solution

    1. $$Integratewithrespecttoxfirst,theny.$$
    2. $$=\frac{27}{4}$$
12. 12.

    Evaluate the iterated integral over the rectangle.

    $${\int }_0^2{\int }_0^1{x}^{2}ydxdy$$

    Answer: $\frac{2}{3}$

    Step-by-step solution

    1. $$Integratewithrespecttoxfirst,theny.$$
    2. $$=\frac{2}{3}$$
13. 13.

    Evaluate the iterated integral over the rectangle.

    $${\int }_0^2{\int }_{-1}^{1}x+2ydxdy$$

    Answer: $8$

    Step-by-step solution

    1. $$Integratewithrespecttoxfirst,theny.$$
    2. $$=8$$
14. 14.

    Evaluate the iterated integral over the rectangle.

    $${\int }_0^1{\int }_0^3{y}^{2}dxdy$$

    Answer: $1$

    Step-by-step solution

    1. $$Integratewithrespecttoxfirst,theny.$$
    2. $$=1$$
15. 15.

    Evaluate the iterated integral over the rectangle.

    $${\int }_0^2{\int }_0^1{x}^{3}dxdy$$

    Answer: $\frac{1}{2}$

    Step-by-step solution

    1. $$Integratewithrespecttoxfirst,theny.$$
    2. $$=\frac{1}{2}$$
16. 16.

    Evaluate the iterated integral over the rectangle.

    $${\int }_0^1{\int }_0^{2}4xydxdy$$

    Answer: $4$

    Step-by-step solution

    1. $$Integratewithrespecttoxfirst,theny.$$
    2. $$=4$$
17. 17.

    Evaluate the iterated integral over the rectangle.

    $${\int }_1^3{\int }_0^1{x}^2+2ydxdy$$

    Answer: $\frac{26}{3}$

    Step-by-step solution

    1. $$Integratewithrespecttoxfirst,theny.$$
    2. $$=\frac{26}{3}$$

More from Multivariable Calculus

• Vectors in 3D
• Lines & Planes
• Partial Derivatives
• Gradients
• Directional Derivatives
• Tangent Planes
• Critical Points
• Polar Integrals
• Triple Integrals
• Divergence & Curl