The area bounded by the 2 parabolas is A(θ) = 1/2∫(r₂²- r₁²).dθ between limits θ = a,b...
<span>the limits are solution to 3cosθ = 1+cosθ the points of intersection of curves. </span>
<span>2cosθ = 1 => θ = ±π/3 </span>
<span>A(θ) = 1/2∫(r₂²- r₁²).dθ = 1/2∫(3cosθ)² - (1+cosθ)².dθ </span>
<span>= 1/2∫(3cosθ)².dθ - 1/2∫(1+cosθ)².dθ </span>
<span>= 9/8[2θ + sin(2θ)] - 1/8[6θ + 8sinθ +sin(2θ)] .. </span>
<span>.............where I have used ∫(cosθ)².dθ=1/4[2θ + sin(2θ)] </span>
<span>= 3θ/2 +sin(2θ) - sin(θ) </span>
<span>Area = A(π/3) - A(-π/3) </span>
<span>= 3π/6 + sin(2π/3) -sin(π/3) - (-3π/6) - sin(-2π/3) + sin(-π/3) </span>
<span>= π.</span>
If you know how to solve word problems involving the sum of consecutive even integers, you should be able to easily solve word problems that involve the sum of consecutive odd integers. The key is to have a good grasp of what odd integers are and how consecutive odd integers can be represented.
Odd Integers
If you recall, an even integer is always 22 times a number. Thus, the general form of an even number is n=2kn=2k, where kk is an integer.
So what does it mean when we say that an integer is odd? Well, it means that it’s one less or one more than an even number. In other words, odd integers are one unit less or one unit more of an even number.
Therefore, the general form of an odd integer can be expressed as nn is n=2k-1n=2k−1 or n=2k+1n=2k+1, where kk is an integer.
Observe that if you’re given an even integer, that even integer is always in between two odd integers. For instance, the even integer 44 is between 33 and 55.
Answer: P = 166 units
Step-by-step explanation: Each point on the triangle has lines that are tangent to the circle in the triangle, meaning that those lines form a 90° angle to the middle point in the circle. There is also a rule that states if there are tangent lines coming from the points and if the points have at least one given side length, the other side will have the same side length. So if one side on point D is 41, then the other side connected to the point is 41. I added up all of the side lengths; 41 + 41 + 30 + 30 + 12 +12 = 166 units.
I hope this helps!