4.4.3Should I sketch rectangles horizontally or vertically?

Multiple Methods for Calculating Area Between Curves

4-155.

Adam, Becky, and Cathy are each working on calculating the area of the region bounded by the curves $y = x^{2}, y = 9$ , and $y = –8x + 9$ . Each person is approaching the problem using a different method, as shown below.
Lesson 4.4.3 Resource Page

Label the dimensions of a typical rectangle in each diagram below.
Describe the technique each student is using. Decide if the method is valid. Then compute each integral to determine if each gives the same area.
A. Adam’s Method
$\text{Area} =\left. \begin{array} { l } { \int _ { 0 } ^ { 1 } ( 9 - ( - 8 x + 9 ) ) d x } \\ { + \int _ { 1 } ^ { 3 } ( 9 - x ^ { 2 } ) d x } \end{array} \right.$
B. Becky’s Method
$\text{Area} =\int _ { 1 } ^ { 9 } ( \sqrt { y } - \frac { 9 - y } { 8 } ) d y$
C. Cathy’s Method
$\text{Area} =\int _ { 1 } ^ { 9 } ( \sqrt { x } - \frac { 9 - x } { 8 } ) d x$

First quadrant, upward parabola vertex at the origin, decreasing line passing through the point (0, comma 9), intersecting parabola at (1, comma 1), horizontal line at, y = 9, intersecting parabola at (3, comma 9), with shaded region below horizontal line, right of slanted line, & inside parabola.

4-156.

Use two different strategies to calculate the area in the first quadrant bounded by $y=\sqrt{x}$ and $y = 4$ . Be prepared to present your solution and describe your method to the class.

4-157.

Calculate the area bound by the curves $y=2\sqrt{x}$ and $y = x$ in the first quadrant by setting up and evaluating an integral expression in terms of $y$ .

4-158.

Calculate the total area enclosed by the functions $y=\sin\left(x\right)$ and $y=-\sin\left(x\right)$ for $0 ≤ x ≤ 2π$ . Be prepared to present your solution and describe your method to the class.

4-159.

FUNKY DESK

The Funky Furniture Company has designed a new desk for schools. The desktop is formed by the region bounded by the functions:

$\left. \begin{array} { l } { f ( x ) = \frac { 1 } { 512 } x ^ { 4 } + \frac { 1 } { 32 } x ^ { 3 } - 2 x - 8 } \\ { g ( x ) = - \frac { 2 } { 225 } x ^ { 2 } + 20 } \end{array} \right.$

The elbowroom is the distance from the $x$ -axis to the lowest point on the curve. How much elbowroom is available on the desk if $x$ and $y$ are measured in inches?
Sketch the region on your paper. Draw and label a typical rectangle that can be used to calculate the area of the desk.
Set up and evaluate an integral to calculate the area of the desktop.

Math Notes

Area Under a Curve vs. Area Between Curves

Definite integrals are often thought of as representing the area of a region on a graph. However, it is important to contrast area between curves with area under a curve.

Area Under a Curve: When evaluating a definite integral, shading the area under a curve is a convenient way to visualize what you are doing. But, in application, definite integrals are rarely used to measure geometric area. Instead, definite integrals are used to calculate the accumulated value of a function.

For example $\int_a^b(\frac{\text{ words }}{\text{ hour }})dt=$ total words read over the interval $[a, b]$

Area Between Curves: However, area between curves always represents a geometric area, measured in square units. For that reason, the integrand must involve the difference between two functions that bound (or contain) a region, and must be set up to guarantee a positive result.

For example: Both definite integrals below represent the geometric area of the region $R$ which is bounded by $y = f\left(x\right)$ and $y = g\left(x\right)$ on the interval $[a, b]$ where $f(x)\ge g(x)$ .

$\int_a^b(f(x)-g(x))dx=$ $\text{ area of }R$ or $\int_a^b|g(x)-f(x)|dx=$ $\text{ area of }R$

Upward parabola, vertex at the point (3, comma negative 1), intersecting a deceasing line at the points (0, comma 8), & (5, comma 3), region below line & above parabola, is shaded.

$\int _ { 0 } ^ { 5 } ( f ( x ) - g ( x ) ) d x$

Upward parabola, vertex at the point (3, comma negative 1), intersecting a decreasing line at the points (0, comma 8), & (5, comma 3), region below line, above parabola, & between x = 1 & x = 4 is shaded.

$\int _ { 1 } ^ { 4 } ( f ( x ) - g ( x ) ) d x$

As shown on the graphs above, the bounds, $x = a$ and $x = b$ , can be the intersection of $y = f\left(x\right)$ and $y = g\left(x\right)$ or (vertical) boundary lines of the region. Note: Whether you are analyzing the area under a curve or computing the area between curves, definite integrals can be set up with reference to the $x$ - or $y$ -axis.

4-160.

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

$\int _ { - 1 } ^ { 1 } \sqrt { x ^ { 2 } } d x$
$\int ( 8 x ^ { 3 } - \frac { 1 } { 2 } x ) d x$
$\int _ { 1 } ^ { 5 } \frac { 3 x ^ { 2 } - 5 x - 2 } { 3 x + 1 } d x$
$\int [ \frac { d } { d x } ( y ) ] d x$

4-161.

Define possible functions $f$ and $g$ so that $h(x) = f(g(x))$ . (Note: $f(x) ≠ x$ and $g(x) ≠ x$ ) Homework Help ✎

$h ( x ) = \sqrt [ 5 ] { \operatorname { cos } ( x ) }$
$h(x)=(3x\cos(x^2))^3$
$h(x) = 1$
$h(x) = x$

4-162.

Determine the value of $a$ such that $f$ is continuous at $x = 3$ .

$f ( x ) = \left\{ \begin{array} { c c } { | 4 - 3 x | } & { \text { for } x < 3 } \\ { a x ^ { 2 } + 2 } & { \text { for } x \geq 3 } \end{array} \right.$ Homework Help ✎

4-163.

Determine the linearization of $y = 4 – 2x^{2}$ at $\left(1, 2\right)$ . Then use it to approximate the value of $y$ when $x = 1.15$ . Homework Help ✎

4-164.

Set up an integral and compute the area of the region bounded by the graphs of the given functions. You may use a graphing calculator. Homework Help ✎

Calculator OK

The area between $y=\sin\left(x\right)$ and $y = \left(x – 1\right)^{4} – 1$ .
The area between $y = x\left(x – 3\right)$ and $y=\sqrt{x}$ .

4-165.

Theresa loves tangents! This time, she has drawn several tangents and erased her original function. What is the equation of her original function? Homework Help ✎

7 segments with approximate slopes at centers as follows: @ (negative 3, comma 8) with slope of negative 4, @ (negative 2, comma 3), slope of negative 2, at (negative 1, comma 0), slope of negative 1, at (0, comma negative 1), slope of 0, at (1, comma 0), slope of 1, at (2, comma 3), slope of 2, at (3, comma 8), slope of 3.

4-166.

For each of the following functions, write an equation for the end-behavior function.
Explain your method. Homework Help ✎

$y = \frac { 6 } { x } + 2 x ^ { 2 }$
$y = \frac { \operatorname { sin } ( x ) } { x }$
$y = \frac { x ^ { 2 } - 3 x - 10 } { x ^ { 2 } + 1 }$

4-167.

For the functions listed in problem 4-166, calculate $\lim\limits _ { x \rightarrow \infty } y$ . Describe the connection between the limit and the end behavior. Homework Help ✎

Calculus Third Edition

Area Under a Curve vs. Area Between Curves