11.1.3Can I calculate the area bounded by two polar curves?

Area Between Polar Curves

11-27.

Halbert enjoys puzzling Judy by making her do problems backwards. He shows Judy the integral
$2 \int _ { 0 } ^ { \pi / 3 } \frac { 1 } { 2 } ( 4 \operatorname { cos } ( \theta ) ) ^ { 2 } d \theta$ ,

and asks her what region this integral calculates the area of. Judy smiles and draws a graph of the region, to which Halbert exclaims, “Right on!” Now it is your turn to draw the region—go for it!

11-28.

Graph the curves $r = 3$ and $r = 6\cos\left(θ\right)$ on the same axes. Shade and calculate the area of the region that is inside of $r = 6\cos\left(θ\right)$ but outside of $r = 3$ .

11-29.

Walker must not have slept enough last night. He is trying to calculate the area from problem 11-28 using a single integral, but he is not sure which of the following integrals is correct:

$\int _ { - \pi / 3 } ^ { \pi / 3 } \frac { 1 } { 2 } ( 6 \operatorname { cos } ( \theta ) - 3 ) ^ { 2 } d \theta$ or $\int _ { - \pi / 3 } ^ { \pi / 3 } \frac { 1 } { 2 } [ ( 6 \operatorname { cos } ( \theta ) ) ^ { 2 } - 3 ^ { 2 } ] d \theta$

Discuss which integral is correct (or are both correct?) with your team. Then write an explanation for Walker about why each integral is correct or not.

11-30.

Consider the following cardioid and circle:

Cardioid: $r = 2 + 2\cos\left(θ\right)$

Circle: $r = 5\cos\left(θ\right)$

For what values of $θ$ do the two curves intersect for $–π ≤ θ ≤ π$ ?
Calculate the area inside the circle but outside the cardioid.
Which is greater: the area inside the circle but outside the cardioid or the area inside the cardioid but outside the circle? Calculate each area and compare.

Circle center @ (2.5, comma 0), with vertices at (2.5, comma 2.5), the origin (5, comma 0), & (2.5, comma negative 2.5), & cardioid with approximate turning points @ origin, (negative 0.5, comma 1), (1.5, comma 2.5), (4, comma 0), (1.5, comma negative 2.5), (negative 0.5, comma negative 1).

11-31.

The position of a ball at time $t$ is specified by the point $(x(t),y(t))$ where the horizontal position is given by $x(t)=20t$ and the vertical position is given by $y(t) = 40t - 10t^2$ . If distance is measured in meters and time is in seconds, Homework Help ✎

When will the ball land?
How far does the ball travel horizontally?
How high does the ball go?

11-32.

Calculate the area within the curve $r = 6 + 3\cos(5θ)$ . Homework Help ✎

11-33.

Examine the integrals below. Consider the multiple tools available for integrating and use the best strategy for each part. Evaluate each integral and briefly describe your method. Homework Help ✎

$\int _ { 2 } ^ { 6 } \frac { 1 } { \sqrt { x - 2 } } d x$

$\int \ln(x)dx$

$\int _ { 3 } ^ { t ^ { 2 } } \frac { d } { d x } ( \frac { x } { x - 2 } ) d x$

$\int _ { 1 } ^ { 2 } \frac { 1 } { 2 x ( x - 3 ) } d x$

11-34.

Use the vectors drawn at right to complete the parts below. Homework Help ✎

Rewrite each vector in both component and $\mathbf{i, j}$ form.
Determine each vector’s standard angle.
What are $\mathbf{\vec { A }}+\mathbf{\vec { B }}$ and $\mathbf{\vec { B }}-\mathbf{\vec { A }}$ ?
Determine $| | \mathbf{\vec { B } }| | \cdot \mathbf{\vec { A }}$ .

11-35.

If $f(x) = g(h(x))$ , then: Homework Help ✎

Explain what each of these expressions represents: $\frac { d g } { d x }$ , $\frac { d g } { d h }$ , and $\frac { d h } { d x }$ .
Explain why $\frac { d g } { d x }=\frac { d g } { d h }\cdot \frac { d h } { d x }$ .
Solve the identity given in part (b) for $\frac { d g } { d h }$ .

11-36.

Multiple Choice: The graph of $1 + x + \frac { x ^ { 2 } } { 2 ! } + \frac { x ^ { 3 } } { 3 ! } + \frac { x ^ { 4 } } { 4 ! } +$ best approximates which function near $x = 0$ ? Homework Help ✎

$f(x) = x^2$

$f\left(x\right) = e^{x}$

$f(x) = \sin(x)$

$f(x) = \cos(x)$

11-37.

Multiple Choice: The cubic $y = x^{3} – ax^{2} + 2$ is concave up on the interval: Homework Help ✎

$\left(0, a\right)$

$( - \infty , \frac { a } { 3 } )$

$( \frac { a } { 3 } , a )$

$( \frac { a } { 3 } , \infty )$

$(−∞, ∞)$

11-38.

Multiple Choice: A racer with a $5$ -foot head start runs with an acceleration of $a(t) = 6t \text{ ft} / \sec^2$ . At $t = 4$ seconds, her velocity is $50$ ft/sec and she finishes the race in $9$ seconds. How long was the race? Homework Help ✎

$734$ ft

$737$ ft

$747$ ft

$750$ ft

$752$ ft

11-39.

Multiple Choice: A point travels along the curve $y = \sin^{–1}(x)$ . If $\frac { d x } { d t }=4$ what is $\frac { d y } { d t }$ when $x =\frac { 1 } { 2 }$ ? Homework Help ✎

$\frac { 2 \sqrt { 3 } } { 3 }$

$\frac { 8 \sqrt { 3 } } { 3 }$

$\frac { 4 \pi } { 3 }$

$\frac { 2 } { 3 }$

$\frac { 8 } { 3 }$

Calculus Third Edition