객관식
Which of the following formulas correctly gives the perimeter of a parallelogram with side lengths and ?
111
views
7. Non-Right Triangles
Area of SAS & ASA Triangles
객관식
Given the polar curve , what is the area enclosed by one complete loop of the curve?
A
B
C
D
검증된 단계별 안내1
Recall that the area enclosed by a polar curve \( r = f(\theta) \) from \( \theta = a \) to \( \theta = b \) is given by the integral formula:
\[ \text{Area} = \frac{1}{2} \int_a^b r^2 \, d\theta \]
Identify the given polar curve:
\[ r = 4 + 3 \sin(\theta) \]
Determine the interval \( [a, b] \) for one complete loop of the curve. Since \( r = 4 + 3 \sin(\theta) \) is a sinusoidal function with period \( 2\pi \), one complete loop corresponds to \( \theta \) going from 0 to \( 2\pi \).
Set up the integral for the area using the formula:
\[ \text{Area} = \frac{1}{2} \int_0^{2\pi} (4 + 3 \sin(\theta))^2 \, d\theta \]
Expand the square inside the integral and simplify the integrand:
\[ (4 + 3 \sin(\theta))^2 = 16 + 24 \sin(\theta) + 9 \sin^2(\theta) \]
Then, integrate each term separately over \( 0 \) to \( 2\pi \) using known integral formulas for \( \sin(\theta) \) and \( \sin^2(\theta) \).
관련 영상
관련 실천
객관식
If the sum of the interior angles of a polygon is , how many sides does the polygon have?
105
views
객관식
Given the polar curves and , what is the area of the region that lies inside both curves?
106
views
