Polar Integrals — Practice Problems and Worked Solutions

The formulas you need here

16 worked problems

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

1. 1.

   Evaluate over the disk using polar coordinates.

   $${\iint }_{x^2+y^2\le 1}(x^2+y^2{)}^{1}dA$$

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

   Step-by-step solution

   1. $$x^2+y^2=r^2,dA=rdrd\theta$$
   2. $$=\frac{\pi }{2}$$
2. 2.

   Evaluate over the disk using polar coordinates.

   $${\iint }_{x^2+y^2\le 4}(x^2+y^2{)}^{1}dA$$

   Answer: $8\pi$

   Step-by-step solution

   1. $$x^2+y^2=r^2,dA=rdrd\theta$$
   2. $$=8\pi$$
3. 3.

   Evaluate over the disk using polar coordinates.

   $${\iint }_{x^2+y^2\le 1}(x^2+y^2{)}^{2}dA$$

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

   Step-by-step solution

   1. $$x^2+y^2=r^2,dA=rdrd\theta$$
   2. $$=\frac{\pi }{3}$$
4. 4.

   Evaluate over the disk using polar coordinates.

   $${\iint }_{x^2+y^2\le 9}(x^2+y^2{)}^{1}dA$$

   Answer: $\frac{81\pi }{2}$

   Step-by-step solution

   1. $$x^2+y^2=r^2,dA=rdrd\theta$$
   2. $$=\frac{81\pi }{2}$$
5. 5.

   Evaluate over the disk using polar coordinates.

   $${\iint }_{x^2+y^2\le 4}(x^2+y^2{)}^{2}dA$$

   Answer: $\frac{64\pi }{3}$

   Step-by-step solution

   1. $$x^2+y^2=r^2,dA=rdrd\theta$$
   2. $$=\frac{64\pi }{3}$$
6. 6.

   Evaluate over the disk using polar coordinates.

   $${\iint }_{x^2+y^2\le 9}(x^2+y^2{)}^{2}dA$$

   Answer: $243\pi$

   Step-by-step solution

   1. $$x^2+y^2=r^2,dA=rdrd\theta$$
   2. $$=243\pi$$
7. 7.

   Evaluate over the disk using polar coordinates.

   $${\iint }_{x^2+y^2\le 1}(x^2+y^2{)}^{3}dA$$

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

   Step-by-step solution

   1. $$x^2+y^2=r^2,dA=rdrd\theta$$
   2. $$=\frac{\pi }{4}$$
8. 8.

   Evaluate over the disk using polar coordinates.

   $${\iint }_{x^2+y^2\le 4}(x^2+y^2{)}^{3}dA$$

   Answer: $64\pi$

   Step-by-step solution

   1. $$x^2+y^2=r^2,dA=rdrd\theta$$
   2. $$=64\pi$$
9. 9.

   Evaluate over the disk using polar coordinates.

   $${\iint }_{x^2+y^2\le 16}(x^2+y^2{)}^{1}dA$$

   Answer: $128\pi$

   Step-by-step solution

   1. $$x^2+y^2=r^2,dA=rdrd\theta$$
   2. $$=128\pi$$
10. 10.

    Evaluate over the disk using polar coordinates.

    $${\iint }_{x^2+y^2\le 16}(x^2+y^2{)}^{2}dA$$

    Answer: $\frac{4096\pi }{3}$

    Step-by-step solution

    1. $$x^2+y^2=r^2,dA=rdrd\theta$$
    2. $$=\frac{4096\pi }{3}$$
11. 11.

    Evaluate over the disk using polar coordinates.

    $${\iint }_{x^2+y^2\le 1}(x^2+y^2{)}^{4}dA$$

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

    Step-by-step solution

    1. $$x^2+y^2=r^2,dA=rdrd\theta$$
    2. $$=\frac{\pi }{5}$$
12. 12.

    Evaluate over the disk using polar coordinates.

    $${\iint }_{x^2+y^2\le 25}(x^2+y^2{)}^{1}dA$$

    Answer: $\frac{625\pi }{2}$

    Step-by-step solution

    1. $$x^2+y^2=r^2,dA=rdrd\theta$$
    2. $$=\frac{625\pi }{2}$$
13. 13.

    Evaluate over the disk using polar coordinates.

    $${\iint }_{x^2+y^2\le 9}(x^2+y^2{)}^{3}dA$$

    Answer: $\frac{6561\pi }{4}$

    Step-by-step solution

    1. $$x^2+y^2=r^2,dA=rdrd\theta$$
    2. $$=\frac{6561\pi }{4}$$
14. 14.

    Evaluate over the disk using polar coordinates.

    $${\iint }_{x^2+y^2\le 4}(x^2+y^2{)}^{4}dA$$

    Answer: $\frac{1024\pi }{5}$

    Step-by-step solution

    1. $$x^2+y^2=r^2,dA=rdrd\theta$$
    2. $$=\frac{1024\pi }{5}$$
15. 15.

    Evaluate over the disk using polar coordinates.

    $${\iint }_{x^2+y^2\le 25}(x^2+y^2{)}^{2}dA$$

    Answer: $\frac{15625\pi }{3}$

    Step-by-step solution

    1. $$x^2+y^2=r^2,dA=rdrd\theta$$
    2. $$=\frac{15625\pi }{3}$$
16. 16.

    Evaluate over the disk using polar coordinates.

    $${\iint }_{x^2+y^2\le 1}(x^2+y^2{)}^{5}dA$$

    Answer: $\frac{\pi }{6}$

    Step-by-step solution

    1. $$x^2+y^2=r^2,dA=rdrd\theta$$
    2. $$=\frac{\pi }{6}$$

More from Multivariable Calculus

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