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###### \${selected_topic_name}
• Notes

Use a double integral to find the area enclosed by one loop of the four-
leaved rose $r=\cos 2 \theta .$

Area $=\int_{y} \int_{x} d x d y=\int_{r_{1}}^{r_{2}} \int_{\theta_{1}}^{\theta_{2}} r d r d \theta$

$\int_{-{\pi/4}}^{\pi/4} \int_{0}^{\cos 2 \theta} r d r d \theta$

$=\int_{-\frac{\pi}{4}}^\frac {\pi}{4}\left[\int_{0}^{\cos 2 \theta} r d r\right] d \theta$

$=\int_{-\frac{\pi}{4}}^{\frac {\pi}{4}} \int_{0}^{\cos 2 \theta} \frac{r^{2}}{2} d\theta$

$=\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{\cos (2 \theta)^{2}}{2}=\frac{\pi}{8}$
The text passage contains a mathematical question related to calculus, specifically about finding the area enclosed by one loop of a four-leaved rose using a double integral. The passage also provides the steps and the final answer to this question.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
• عربي

Need Help?

Subscribe to Calculus C

###### \${selected_topic_name}
• Notes

Use a double integral to find the area enclosed by one loop of the four-
leaved rose $r=\cos 2 \theta .$

Area $=\int_{y} \int_{x} d x d y=\int_{r_{1}}^{r_{2}} \int_{\theta_{1}}^{\theta_{2}} r d r d \theta$

$\int_{-{\pi/4}}^{\pi/4} \int_{0}^{\cos 2 \theta} r d r d \theta$

$=\int_{-\frac{\pi}{4}}^\frac {\pi}{4}\left[\int_{0}^{\cos 2 \theta} r d r\right] d \theta$

$=\int_{-\frac{\pi}{4}}^{\frac {\pi}{4}} \int_{0}^{\cos 2 \theta} \frac{r^{2}}{2} d\theta$

$=\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{\cos (2 \theta)^{2}}{2}=\frac{\pi}{8}$
user
The text passage contains a mathematical question related to calculus, specifically about finding the area enclosed by one loop of a four-leaved rose using a double integral. The passage also provides the steps and the final answer to this question.

TYPE: SINGLE_QUESTION_WITH_ANSWER