Solution Found!
Answer: 5057. Miscellaneous integrals Evaluate the
Chapter 13, Problem 57(choose chapter or problem)
Evaluate the following integrals using the method of your choice. A sketch is helpful.
\(\iint_{R} \frac{d A}{4+\sqrt{x^{2}+y^{2}}} ; R=\{(r, \theta): 0 \leq r \leq 2,\ \pi / 2 \leq \theta \leq 3 \pi / 2\}\)
Questions & Answers
QUESTION:
Evaluate the following integrals using the method of your choice. A sketch is helpful.
\(\iint_{R} \frac{d A}{4+\sqrt{x^{2}+y^{2}}} ; R=\{(r, \theta): 0 \leq r \leq 2,\ \pi / 2 \leq \theta \leq 3 \pi / 2\}\)
ANSWER:Step 1 of 2
\(\iint_{R} \frac{d A}{4+\sqrt{x^{2}+y^{2}}} ; R=\{(r, \theta): 0 \leq r \leq 2,\ \pi / 2 \leq \theta \leq 3 \pi / 2\}\)
therefore, using polar coordinates :
\(=\int_{\pi / 2}^{3 \pi / 2} \int_{0}^{2} \frac{r d r d \theta}{4+r}=\int_{\pi / 2}^{3 \pi / 2} \int_{0}^{2}\left(1-\frac{4}{1+r}\right) d r d \theta\)
integral with respect to r and evaluate
\(\begin{array}{l} =\int_{\pi / 2}^{3 \pi / 2}\left(r-4 \ln \mid 4+r \|_{0}^{2} d \theta\right. \\ =\int_{\pi / 2}^{3 \pi / 2}(2-4(\ln 6-\ln 4)) d \theta=\int_{\pi / 2}^{3 \pi / 2}\left(2-4 \ln \left(\frac{3}{2}\right)\right) d \theta \end{array}\)