How to Find the Area of a Polar Curve with Cosine

TL;DR

To find the area of a shaded region enclosed by a polar curve like R = cos(3θ), integrate from 0 to π instead of 0 to 2π to avoid duplication. Use the formula for area A = 1/2 ∫ (R² dθ) and divide the result by the number of petals (3 for this curve) to get the area of one petal.

Transcript

okay another question on my final example here we are going to find the area of this shaded region and we have a polar curve R is equal to cosine of 3 theta well notice were three and we have one two three three petals right here right the truth is when you have an odd number here you will have the same amount of petals right here for the whole gra... Read More

Key Insights

• #️⃣ The number of petals in a polar curve with a cosine function is determined by the odd number in the function's argument.
• 🤝 Integrating from 0 to PI is sufficient to obtain the entire graph when dealing with an odd number in the cosine function.
• 🐻‍❄️ The formula for finding the area of a polar curve is 1/2 times R squared, and for a shaded region, divide the result by the number of petals.
• 😫 The theta values for the shaded region can be determined by setting the function R equal to zero and solving for theta.

Questions & Answers

Q: How can you determine the number of petals in a polar curve with cosine of 3 theta?

When the cosine function has an odd number in the argument, such as 3 theta, the number of petals in the graph will be the same as the number in the function, which is 3 in this case.

Q: Why is it sufficient to integrate from 0 to PI to obtain the whole graph?

Integrating from 0 to PI is enough because an odd number in the cosine function creates a symmetrical graph, so integrating from 0 to 2PI would duplicate the graph.

Q: What is the formula for finding the area of a polar curve?

The formula is 1/2 times R squared, where R represents the function of theta. To find the area of a shaded region, divide the result by the number of petals.

Q: How can you determine the theta values for the shaded region in the example?

To determine the theta values, set the function R equal to zero. In this case, cosine of 3 theta equals zero, which results in odd multiples of PI over 6 as the theta values for the shaded region.

Summary & Key Takeaways

• When dealing with a polar curve with an odd number in the cosine function, the number of petals in the graph is equal to the number in the function.

• To obtain the whole graph, only integrate from 0 to PI; going from 0 to 2PI would result in duplicating the graph.

• The formula for the area of a polar curve is 1/2 times R squared, with R being the function of theta, and for a shaded region, divide the result by the number of petals.