find the area of the region inside the limacon r= 8 + 4sin(theta)
HElp!! if you can show steps, i'd be greatful
for polar coordinates, dA = 1/2 r^2 dθ since each tiny wedge of area can be considered a sector of a circle of radius r and angle dθ.
So, since
r = 8+4sinθ,
a = ∫[0,2π] 1/2 (8+4sinθ)^2 dθ
now just expand the quadratic and evaluate each term, and you wind up with
a = 72π
Find the area of the specified region.
Inside the limacon r = 8 + 2 sin θ
66 pi
To find the area of the region inside the limaçon, you can use the concept of polar coordinates. Here's how you can find the area step by step:
Step 1: Sketch the graph: Start by sketching the limaçon to visualize the region. The equation of the limaçon is given by r = 8 + 4sin(theta). You can plot points by substituting different values of theta into the equation and locating the corresponding r-values. Connect these points to form the graph of the limaçon.
Step 2: Determine the boundaries of integration: Look for the values of theta at which the limaçon intersects the origin (r = 0). These points will help determine the limits of integration for theta.
To find these points, set r = 0 and solve for theta:
0 = 8 + 4sin(theta)
4sin(theta) = -8
sin(theta) = -2
Since -2 is outside the range [-1, 1], there are no values of theta that satisfy this equation. Therefore, the limaçon does not intersect the origin, and the boundaries of integration for theta are from 0 to 2π (a full revolution).
Step 3: Set up the integral for area: The area within the polar curve can be calculated using the integral formula:
A = (1/2) ∫[θ1 to θ2] (r^2) dθ
In this case, r = 8 + 4sin(theta), so the integral becomes:
A = (1/2) ∫[0 to 2π] ((8 + 4sin(theta))^2) dθ
Step 4: Simplify and solve the integral: Expand the expression inside the integral and simplify it as much as possible. Then integrate with respect to theta.
A = (1/2) ∫[0 to 2π] (64 + 64sin(theta) + 16sin^2(theta)) dθ
To integrate, use trigonometric identities to convert sin^2(theta) into terms that can be integrated easily.
A = (1/2) ∫[0 to 2π] (64 + 64sin(theta) + 16(1-cos^2(theta))) dθ
= (1/2) ∫[0 to 2π] (80 + 64sin(theta) - 16cos^2(theta)) dθ
Now, integrate each term separately. The integral of a constant is the constant multiplied by the range of integration.
A = (1/2) [(80θ) + (64(-cos(theta))) - (16/3)(cos^3(theta))] [0 to 2π]
A = (1/2) [(80(2π)) + (64(-cos(2π))) - (16/3)(cos^3(2π))] - [(80(0)) + (64(-cos(0))) - (16/3)(cos^3(0))]
Simplyfing and calculating further, you should get the final value of the area within the limaçon.
Note: If you prefer a numerical answer, use a calculator to evaluate the integral and arrive at the approximate area.