4.4.2How do I calculate the area bounded by three curves?

More Area Between Curves

4-142.

Calculate the area of the regions described below. Review problem 4-132 for a description of a complete solution.

The area between $y = –2\left(x^{2} – 1\right)$ and $y = –x^{2} + 1$ .
The area between $y=\sin\left(x\right)$ and $y=\frac{3}{4}x−7$ for $π ≤ x ≤ 2π$ .

4-143.

What will the result be if $\int _ { a } ^ { b } ( g ( x ) - f ( x ) ) d x$ is calculated instead of $\int _ { a } ^ { b } ( f ( x ) - g ( x ) ) d x$ for the functions $f$ and $g$ shown below? Explain your thinking.

First quadrant, 2 intersecting curves, concave up labeled, g of x, concave down labeled, f of x, with tick marks on x axis, corresponding to the points of intersection, labeled, a & b, shaded vertical rectangle with top left vertex on, f of x, & bottom left vertex on, g of x, about a third right of left intersection.

4-144.

Examine the area of the region bounded by the graphs of the equations $y = x^{2}$ , $y=\sqrt{x}$ , and $y = –x + 6$ , shown at right.

Explain why you cannot use one integral to calculate the area of the entire region.
Write an expression using two integrals that will calculate the total area of the bounded region using typical rectangles that are vertical. Then, evaluate the integrals and calculate the area.

First quadrant, upward parabola, vertex at the origin, passing through the points (1, comma 1) & (2, comma 4), increasing concave down curve, starting at the origin, passing through the points (1, comma 1), & (4, comma 2), & decreasing line, passing through the points (2, comma 4), & at (4, comma 2).

4-145.

Examine the area bounded by the graphs of the functions $f\left(x\right) = x – 1$ and $g\left(x\right) = x^{3} – 2x^{2} + 1$ , shown at right.

Explain why this region requires two integrals.
Write and evaluate a set of integrals to calculate the area between the curves. Check your answer with your graphing calculator.

4-146.

Examine the integrals below. Consider the multiple tools available for evaluating integrals and use the best strategy for each part. Evaluate the definite integrals and state the strategies that you used. For the indefinite integrals, find the antiderivative function, if you can. Homework Help ✎

$\int \sqrt { 1 - x ^ { 4 } } d x$
$\int ( \frac { 4 } { m ^ { 3 } } - 3 \operatorname { cos } ( m ) ) d m$
$\int _ { 1 } ^ { 2 } x ^ { x } d x$
$\int \pi ^ { 2 } d x$
$\int _ { 0 } ^ { 5 } ( | x - 2 | + 3 ) d x$

4-147.

Draw a flag that will generate the same volume no matter if it is rotated about the $x$ - or $y$ -axis. Is there more than one possible shape of flag that meets this requirement? Does your flag have any special property that ensures these equal volumes of rotation? Homework Help ✎

4-148.

Without your calculator, evaluate the following limits. Homework Help ✎
Compute without a calculator

$\lim\limits _ { x \rightarrow 3 ^ { + } } \sqrt { x - 3 }$
$\lim\limits _ { x \rightarrow \infty } \frac { x ^ { 2 } - 2 x + 1 } { x ^ { 3 } }$
$\lim\limits _ { x \rightarrow \pi } \frac { \operatorname { cos } ( x ) + 1 } { x - \pi }$

4-149.

Use your calculator to approximate the following limits, accurate to three decimal places. Homework Help ✎

$\lim\limits_ { x \rightarrow 1 } \frac { 2 ^ { x } - 2 } { 3 ^ { x } - 3 }$
$\lim\limits _ { x \rightarrow 0 } \frac { 1 - \operatorname { cos } ( x ) } { x ^ { 2 } }$

4-150.

Write the equation of the line that is: Homework Help ✎

Tangent to the function $y = \frac { 3 x ^ { 2 } - 12 } { x + 2 }$ at $x = 1$ .
Perpendicular to the tangent line in part (a) at $x = 1$ .

4-151.

The annual cycle of daily high temperatures in Cabanaville is shown in the graph below from January 1st to December 31st. The $x$ -axis is marked in months and the $y$ -axis represents temperature in $^\circ\text{F}$ . Homework Help ✎

First quadrant, x axis with 12 tick marks, periodic curve starting on the y axis at y = 50, with 2 visible turning points at the first tick at y = 40, & at the seventh tick mark at y = 100.

Approximately when is Cabanaville at its hottest? The coldest? How can you tell?
When is the temperature changing the fastest? What is the name for this type point on
the graph?

4-152.

Determine the value of $a$ such that $f$ is differentiable at $x = 1$ . 4-152 HW eTool Homework Help ✎

$f ( x ) = \left\{ \begin{array} { l l } { ( x + 2 ) ^ { 2 } - 3 } & { \text { for } x < 1 } \\ { a \operatorname { sin } ( x - 1 ) + 6 } & { \text { for } x \geq 1 } \end{array} \right.$

4-153.

The graph of $y = f^\prime\left(x\right)$ is shown at right. Determine the intervals where $f$ is increasing, decreasing, concave up, and concave down. Homework Help ✎

Continuous curve labeled, f prime of x, coming from lower left, turning at the approximate points (negative 1, comma 3), & (3, comma negative 0.5), passing through the x axis at negative 3, 2, & 4, changing from concave down to concave up at about (1, comma 1).

4-154.

If $n$ is a positive integer, write a definite integral to represent $\lim\limits _ { n \rightarrow \infty } \frac { 2 } { n } [ ( 1 + \frac { 2 } { n } ) ^ { 2 } + ( 1 + \frac { 4 } { n } ) ^ { 2 } + \ldots + ( 1 + \frac { 2 n } { n } ) ^ { 2 } ]$ . Homework Help ✎

Calculus Third Edition