11.1.2How can I get the exact area?

More Polar Area

11-15.

The equation for the limaçon in problem 11-2 is $r=7+6\cos(θ)$ . Now we will calculate the exact area bounded by this curve.

State a minimum interval of values for $θ$ that will generate the entire curve.
Write and evaluate an integral expression for the area within the limaçon without using a calculator.
How accurate was the area you calculated in problem 11-2?

Compute without a calculator

11-16.

Graph the rose $r=\cos(2θ)$ for $-\pi\leθ\le\pi$ and sketch the figure on your paper:

For what values of $θ$ does the curve pass through the pole?
Analyze the following integral. Explain what it represents.

$4 \int _ { - \pi / 4 } ^ { \pi / 4 } \frac { 1 } { 2 } \operatorname { cos } ^ { 2 } ( 2 \theta ) d \theta$
Without a calculator, evaluate the integral to calculate the area bounded by $r=\cos(2θ)$ .

Compute without a calculator

11-17.

Several polar curves have special names. One of those is the Lemniscate of Bernoulli, shown at right. The equation for this curve is

$r=\sqrt{36\cos(2\theta)}$

Calculate the total area within lemniscate.

Connected curve, turning at the approximate points, (negative 6, comma 0), (negative 4, comma 2), (0, comma 0), (4, comma negative 2), (6, comma 0), (4, comma 2), (0, comma 0), (negative 4, comma negative 2), back to (negative 6, comma 0).

11-18.

Calculate the area of each of the shaded regions shown below.

$r=\cos(2θ)\text{ and }r=1$

Shaded circle centered at the origin, with 4 unshaded petals, 1 on each axes.

$r = \sec(θ)$

Unscaled axes, vertical line, half way on positive x axis, shaded triangle, vertices on the origin, on positive y values on vertical line, & on negative portion on vertical line, top right vertex labeled, theta = 1 fourth pi, bottom right vertex labeled, theta = negative 1 fourth pi.

11-19.

Calculate the area bounded by $r = 3\sin(3θ)$ .

11-20.

Consider the infinite series below. For each series, decide if it diverges, converges conditionally, or converges absolutely and justify your conclusion. State the tests you used. Homework Help ✎

$\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { - 1 } { n + 2 }$

$\displaystyle\sum _ { n = 1 } ^ { \infty } \operatorname { ln } ( n )$

$\displaystyle\sum _ { n = 1 } ^ { \infty } 0.1$

$\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 1.01 } }$

11-21.

Given $\mathbf{a} = 3\mathbf{i} + 5\mathbf{j},$ $\mathbf{b} = -9\mathbf{j}, \text{ and }$ $\mathbf{c} = 6\mathbf{i} - 8\mathbf{j}$ , what are: Homework Help ✎

$6\mathbf{a} + \mathbf{b}$

$\frac { 1 } { 2 }\mathbf{c + a}$

$5\frac { \mathbf{c} } { \| \mathbf{c} \| }$

$\mathbf{| | a - b | |}$

11-22.

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 ✎

$\int \frac { 1 } { 2 } \operatorname { sec } ^ { 2 } ( x ) \operatorname { tan } ( x ) d x$

$\int \frac { \operatorname { cos } ^ { 2 } ( x ) - \operatorname { sin } ^ { 2 } ( x ) } { \operatorname { sin } ( 2 x ) } d x$

$\int _ { 0 } ^ { a } ( a x ^ { 2 / 3 } - b ) d x$

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

11-23.

Multiple Choice: $\lim\limits _ { x \rightarrow 0 } \frac { x - 1 + \sqrt { x + 1 } } { x }$ = Homework Help ✎

$\frac { 2 } { 3 }$

$1.5$

11-24.

Multiple Choice: The function $f(x) =|x ^ { 2 / 3 } - 1|+2$ in non-differentiable at $x =$ : 11-24 HW eTool (Desmos). Homework Help ✎

$-1$

$-1, 1$

$-1, 0, 1$

11-25.

Multiple Choice: $\frac { d } { d x } \int _ { 2 x } ^ { x ^ { 2 } } ( m ^ { 2 } - 4 m ) d m$ = Homework Help ✎

$\frac { 1 } { 3 }x^6 − 2x^4$

$\frac { 1 } { 3 }x^6 − 2x^4 + C$

$x^2 − 4x$

$2x^3 − 9x^2 + 4x$

$2x^5 − 8x^3 − 8x^2 + 16x$

11-26.

Multiple Choice: A $4$ -foot tall child walks away from a $12$ -foot tall street light at a rate of $2$ ft/sec. At what rate is the length of her shadow increasing when she is $6$ feet away from the light? Homework Help ✎

$1$ ft/sec

$2$ ft/sec

$3$ ft/sec

$4$ ft/sec

$6$ ft/sec

Calculus Third Edition

11.1.2How can I get the exact area?

Area for Polar Graphs