Double integrals in polar coordinates practice problems
Disks and circles: dA = r dr dθ. 12 problems with worked solutions; in 12 of them every equation is proved by a computer algebra system.
Evaluate \( \displaystyle \iint_D \sqrt{x^{2} + y^{2}} \, dA \) over the disk \( \displaystyle x^2 + y^2 \le 1 \) using polar coordinates.
Evaluate \( \displaystyle \iint_D \sqrt{x^{2} + y^{2}} \, dA \) over the disk \( \displaystyle x^2 + y^2 \le 9 \) using polar coordinates.
Evaluate \( \displaystyle \iint_D x^{2} + y^{2} + 1 \, dA \) over the disk \( \displaystyle x^2 + y^2 \le 4 \) using polar coordinates.
Evaluate \( \displaystyle \iint_D x^{2} + y^{2} \, dA \) over the disk \( \displaystyle x^2 + y^2 \le 4 \) using polar coordinates.
Evaluate \( \displaystyle \iint_D x^{2} + y^{2} \, dA \) over the disk \( \displaystyle x^2 + y^2 \le 1 \) using polar coordinates.
Evaluate \( \displaystyle \iint_D x^{2} + y^{2} + 1 \, dA \) over the disk \( \displaystyle x^2 + y^2 \le 9 \) using polar coordinates.
Evaluate \( \displaystyle \iint_D x^{2} + y^{2} \, dA \) over the disk \( \displaystyle x^2 + y^2 \le 9 \) using polar coordinates.
Evaluate \( \displaystyle \iint_D x^{2} + y^{2} + 1 \, dA \) over the disk \( \displaystyle x^2 + y^2 \le 1 \) using polar coordinates.
Evaluate \( \displaystyle \iint_D \sqrt{x^{2} + y^{2}} \, dA \) over the disk \( \displaystyle x^2 + y^2 \le 4 \) using polar coordinates.
Evaluate \( \displaystyle \iint_D e^{- x^{2} - y^{2}} \, dA \) over the disk \( \displaystyle x^2 + y^2 \le 1 \) using polar coordinates.
Evaluate \( \displaystyle \iint_D e^{- x^{2} - y^{2}} \, dA \) over the disk \( \displaystyle x^2 + y^2 \le 4 \) using polar coordinates.
Evaluate \( \displaystyle \iint_D e^{- x^{2} - y^{2}} \, dA \) over the disk \( \displaystyle x^2 + y^2 \le 9 \) using polar coordinates.