AP Calculus BC Polar Curves Quiz

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Questions and Answers

What is the area of the region R bounded by the polar curve in the first quadrant?

2.618
0.465
0.317
0.929 (correct)

In the figure above, what is the area of the shaded region formed by a polar curve?

Option C
Option A
Option B (correct)
Option D

For the polar curves in the graph above, which integral expression gives the area of the region bounded between them?

∭
∬
∮ (correct)
∯

What is the area of the region bounded by the two polar curves shown in the figure above?

26.704 (B) Signup and view all the answers

In the graph above, two polar curves intersect at a point. What is the area of the shaded region?

2.040 (C) Signup and view all the answers

For the given region in the first and second quadrants, which integral expression provides the area between the specified polar curves?

∬ (C) Signup and view all the answers

Which of the following must be true if $\int_a^\infty f(x) dx$ converges?

$f(x)$ is decreasing (A) Signup and view all the answers

If a series converges to $k$, what must be true about the series?

It converges (D) Signup and view all the answers

Which of the following inequalities is true for verifying convergence using the integral test?

$\frac{d}{dx} f(x) > 0$ (B) Signup and view all the answers

If $\sum_{n=1}^{\infty} a_n$ converges, what must be true about the terms of the series?

They have limit 0 (B) Signup and view all the answers

Which of the following statements is true about a series that diverges and has terms with a limit of 0?

The series diverges (A) Signup and view all the answers

$\int_a^b f(x) dx$ converges. Which of the following must be true about $f(x)$?

$f(x)$ is positive (D) Signup and view all the answers

What is the total area between the polar curves $r = 5 imes ext{sin}(3 heta)$ and $r = 8 imes ext{sin}(3 heta)$?

61.261 (B) Signup and view all the answers

Which of the following integrals represents the area enclosed by the smaller loop of the graph of the polar curve?

$\int_{\pi/3}^{\pi/2} r^2 d\theta$ (C) Signup and view all the answers

What is the area of the region in the first quadrant bounded by $r = \cos(\theta)$ and $r = 2$ when they intersect at $\theta = 0.450$?

0.271 (A) Signup and view all the answers

Which of the following integrals gives the area of the region bounded by $r = 3 + 2 \cos(\theta)$, $r = 1$, and $r = 4 \sin(\theta)$?

$\int_{0}^{2\pi} (4 \sin(\theta) - 3 - 2 \cos(\theta)) d\theta$ (C) Signup and view all the answers

Which of the following expressions gives the total area of the shaded regions within the polar curves shown in the figure?

$r_1 \times r_2$ (D) Signup and view all the answers

What is the sum of the areas enclosed by the shaded regions of the polar curves shown in the figure?

9.425 (B) Signup and view all the answers

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Study Notes

Polar Curves and Area

A polar curve is given by a differentiable function, and the line tangent to the curve at a point is horizontal.
For a certain polar curve, the value of r at θ = π/4 is 0.417, and dr/dθ at θ = π/4 is 3.195.

Area Between Polar Curves

The total area between the polar curves r = 5 sin(3θ) and r = 8 sin(3θ) is 30.631.
The integral ∫(1/2)r^2 dθ represents the area enclosed by a polar curve.
The area of the region bounded by the graphs of the polar curves r = cos(θ) and r = 2 is 0.243.

Area of Regions

The area of the region bounded by the graphs of the polar curves r = 2 + cos(θ) and r = 2 is 3.142.
The area of the region bounded by the graphs of the polar curves r = 2 and r = 4 sin(θ) is 0.858.
The area of the region bounded by the graphs of the polar curves r = 1 and r = 2 sin(θ) is 0.465.

Integral Expressions

The integral ∫(1/2)r^2 dθ gives the area of the region bounded by a polar curve.
The integral ∫(1/2)(r1^2 - r2^2) dθ gives the area between two polar curves r1 and r2.

Series and Convergence

If a series converges and an → 0 as n → ∞, then ∑an converges.
The integral test can be used to determine the convergence of a series.
If ∫[0,∞) f(x) dx converges, then ∑[0,∞) f(n) converges.

Convergence of Series

If f is a positive, continuous, decreasing function and ∑f(n) converges, then ∫[0,∞) f(x) dx converges.
If f is a positive, continuous, decreasing function and ∫[0,∞) f(x) dx converges, then ∑f(n) converges.
The integral test can be used to determine the convergence of a series.

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