- #1
Huski
Hi, everyone. I had an example from my book, but I wasn't sure how they got [itex]\dfrac{1}{2}cos\theta[/itex] on step 7? It seems like once they combined the constants, they ended up with just [itex]cos2\theta[/itex]. Although, they have a [itex]\dfrac{1}{2}[/itex] in front. Can someone help me understand where that constant came from? Thank you.
1. Homework Statement
Find the area of the region enclosed by the polar curve [itex]r=3-3cos\theta[/itex]
Area in Polar Coordinates
[itex]dA=\dfrac{1}{2}r^2d\theta[/itex]
Double Angle Identity:
[itex]cos2\theta=2cos^2\theta-1[/itex]
1. [itex]dA=\dfrac{1}{2}(3-3cos\theta)^2d\theta[/itex]
factor
2. [itex]dA=\dfrac{1}{2}(9-18cos\theta+9cos^2\theta)d\theta[/itex]
write the integral and factor [itex]\dfrac{1}{2}[/itex] through
3. [itex]\displaystyle A=\dfrac{1}{2}\int_{0}^{2\pi}(9-18cos^2\theta+9cos^2\theta)d\theta[/itex]
factor out 9
4. [itex]\displaystyle A=9\cdot\dfrac{1}{2}\int_{0}^{2\pi}(1-2cos\theta+cos^2\theta)d\theta[/itex]
rewrite double-angle identity
5. [itex]cos^2\theta=\dfrac{cos^2\theta+1}{2}[/itex]
replace [itex]cos^2\theta[/itex] with the double-angle identity
6. [itex]\displaystyle A=\dfrac{9}{2}\int_{0}^{2\pi}\left(1-2cos\theta+\dfrac{cos^2\theta+1}{2} \right)d\theta[/itex]
combine the constants to get [itex]\dfrac{3}{2}[/itex]
7. [itex]\displaystyle A=\dfrac{9}{2}\int_{0}^{2\pi}\left(\dfrac{3}{2}-2cos\theta+\dfrac{1}{2}cos2\theta \right)d\theta[/itex]
integrate
8. [itex]\dfrac{9}{2}\left[\dfrac{3}{2}\theta-2sin\theta+\dfrac{1}{4}sin\theta \right]_{0}^{2\pi}[/itex]
answer
9. [itex]\dfrac{27\pi}{2}[/itex]
1. Homework Statement
Find the area of the region enclosed by the polar curve [itex]r=3-3cos\theta[/itex]
Homework Equations
Area in Polar Coordinates
[itex]dA=\dfrac{1}{2}r^2d\theta[/itex]
Double Angle Identity:
[itex]cos2\theta=2cos^2\theta-1[/itex]
The Attempt at a Solution
1. [itex]dA=\dfrac{1}{2}(3-3cos\theta)^2d\theta[/itex]
factor
2. [itex]dA=\dfrac{1}{2}(9-18cos\theta+9cos^2\theta)d\theta[/itex]
write the integral and factor [itex]\dfrac{1}{2}[/itex] through
3. [itex]\displaystyle A=\dfrac{1}{2}\int_{0}^{2\pi}(9-18cos^2\theta+9cos^2\theta)d\theta[/itex]
factor out 9
4. [itex]\displaystyle A=9\cdot\dfrac{1}{2}\int_{0}^{2\pi}(1-2cos\theta+cos^2\theta)d\theta[/itex]
rewrite double-angle identity
5. [itex]cos^2\theta=\dfrac{cos^2\theta+1}{2}[/itex]
replace [itex]cos^2\theta[/itex] with the double-angle identity
6. [itex]\displaystyle A=\dfrac{9}{2}\int_{0}^{2\pi}\left(1-2cos\theta+\dfrac{cos^2\theta+1}{2} \right)d\theta[/itex]
combine the constants to get [itex]\dfrac{3}{2}[/itex]
7. [itex]\displaystyle A=\dfrac{9}{2}\int_{0}^{2\pi}\left(\dfrac{3}{2}-2cos\theta+\dfrac{1}{2}cos2\theta \right)d\theta[/itex]
integrate
8. [itex]\dfrac{9}{2}\left[\dfrac{3}{2}\theta-2sin\theta+\dfrac{1}{4}sin\theta \right]_{0}^{2\pi}[/itex]
answer
9. [itex]\dfrac{27\pi}{2}[/itex]