Double integrals over general regions practice problems
Type I and Type II regions: set the bounds, then integrate. 30 problems with worked solutions; in 30 of them every equation is proved by a computer algebra system.
Evaluate \( \displaystyle \iint_D x^{2} \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = x \), \( \displaystyle x = 0 \) and \( \displaystyle x = 1 \).
Evaluate \( \displaystyle \iint_D x + y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = x \), \( \displaystyle x = 0 \) and \( \displaystyle x = 2 \).
Evaluate \( \displaystyle \iint_D 2 x \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = 2 x \), \( \displaystyle x = 0 \) and \( \displaystyle x = 1 \).
Evaluate \( \displaystyle \iint_D x y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = x \), \( \displaystyle x = 0 \) and \( \displaystyle x = 1 \).
Evaluate \( \displaystyle \iint_D 2 x \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = x^{2} \), \( \displaystyle x = 0 \) and \( \displaystyle x = 2 \).
Evaluate \( \displaystyle \iint_D 6 x y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = x^{2} \), \( \displaystyle x = 0 \) and \( \displaystyle x = 2 \).
Evaluate \( \displaystyle \iint_D x^{2} \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = \sqrt{x} \), \( \displaystyle x = 0 \) and \( \displaystyle x = 1 \).
Evaluate \( \displaystyle \iint_D x y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = 2 x \), \( \displaystyle x = 0 \) and \( \displaystyle x = 2 \).
Evaluate \( \displaystyle \iint_D x y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = \sqrt{x} \), \( \displaystyle x = 0 \) and \( \displaystyle x = 1 \).
Evaluate \( \displaystyle \iint_D 6 x y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = x^{2} \), \( \displaystyle x = 0 \) and \( \displaystyle x = 1 \).
Evaluate \( \displaystyle \iint_D 6 x y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = x \), \( \displaystyle x = 0 \) and \( \displaystyle x = 1 \).
Evaluate \( \displaystyle \iint_D 2 x \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = x \), \( \displaystyle x = 0 \) and \( \displaystyle x = 2 \).
Evaluate \( \displaystyle \iint_D x^{2} \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = 2 x \), \( \displaystyle x = 0 \) and \( \displaystyle x = 1 \).
Evaluate \( \displaystyle \iint_D 2 x \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = x \), \( \displaystyle x = 0 \) and \( \displaystyle x = 1 \).
Evaluate \( \displaystyle \iint_D y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = x \), \( \displaystyle x = 0 \) and \( \displaystyle x = 1 \).
Evaluate \( \displaystyle \iint_D x y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = x \), \( \displaystyle x = 0 \) and \( \displaystyle x = 2 \).
Evaluate \( \displaystyle \iint_D 2 x \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = 2 x \), \( \displaystyle x = 0 \) and \( \displaystyle x = 2 \).
Evaluate \( \displaystyle \iint_D 6 x y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = 2 x \), \( \displaystyle x = 0 \) and \( \displaystyle x = 2 \).
Evaluate \( \displaystyle \iint_D 6 x y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = \sqrt{x} \), \( \displaystyle x = 0 \) and \( \displaystyle x = 1 \).
Evaluate \( \displaystyle \iint_D 6 x y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = x \), \( \displaystyle x = 0 \) and \( \displaystyle x = 2 \).
Evaluate \( \displaystyle \iint_D x y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = x^{2} \), \( \displaystyle x = 0 \) and \( \displaystyle x = 1 \).
Evaluate \( \displaystyle \iint_D 6 x y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = 2 x \), \( \displaystyle x = 0 \) and \( \displaystyle x = 1 \).
Evaluate \( \displaystyle \iint_D y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = \sqrt{x} \), \( \displaystyle x = 0 \) and \( \displaystyle x = 2 \).
Evaluate \( \displaystyle \iint_D x^{2} \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = x \), \( \displaystyle x = 0 \) and \( \displaystyle x = 2 \).
Evaluate \( \displaystyle \iint_D y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = \sqrt{x} \), \( \displaystyle x = 0 \) and \( \displaystyle x = 1 \).
Evaluate \( \displaystyle \iint_D x y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = x^{2} \), \( \displaystyle x = 0 \) and \( \displaystyle x = 2 \).
Evaluate \( \displaystyle \iint_D y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = 2 x \), \( \displaystyle x = 0 \) and \( \displaystyle x = 2 \).
Evaluate \( \displaystyle \iint_D y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = 2 x \), \( \displaystyle x = 0 \) and \( \displaystyle x = 1 \).
Evaluate \( \displaystyle \iint_D x^{2} \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = \sqrt{x} \), \( \displaystyle x = 0 \) and \( \displaystyle x = 2 \).
Evaluate \( \displaystyle \iint_D x + y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = \sqrt{x} \), \( \displaystyle x = 0 \) and \( \displaystyle x = 2 \).