Area Between CurvesPrerequisite KnowledgeAlgebraTheorem: Fundamental Theorem of Algebra [FTA]Definition: DiscriminantDefinition: Quadratic FormulaSolving EquationsFind Intersections of Two CurvesFind Intercepts of Curves

**Author**: John J Weber III, PhD
**Corresponding Textbook Sections**:

**Section 6.1**– Area Between Curves

Every polynomial of degree

**NOTE**: We will only need to find the real roots of equations.

The **discriminant** to the quadratic equation

To find the solutions to

If the discriminant is non-negative, i.e.,

Quadratic functions:

- use Quadratic Formula or use method below for Polynomial functions

Polynomial functions:

- Move all terms to one side of the equation:
${a}_{n}{x}^{n}+{a}_{n-1}{x}^{n-1}+\cdots +{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}=0$ - Factor into linear factors, if possible:
$(x-b)(x-c)\cdots (x-p)=0$ - Use the
**Zero Product Property of Real Numbers**, i.e., set each factor to and solve for$0$ .$x$ - Use FTA to check that you found all real solutions.

- Move all terms to one side of the equation:
Root functions:

, when$\sqrt{\text{arg}(x)}=0$ , for any function,$\text{arg}(x)=0$ .$\text{arg}(x)$

Exponential functions:

- Exponential functions cannot be equal to
unless the function is shifted vertically.$0$

- Exponential functions cannot be equal to
Logarithmic functions:

, when${\mathrm{log}}_{b}\left(\text{arg}(x)\right)=0$ , for any function,$\text{arg}(x)=1$ .$\text{arg}(x)$

Trigonometric functions:

- Use the unit circle to determine when a trig function is zero.

Suppose,

Suppose,

Replace all

Replace all

**NOTE**: *Sketching* is **not** the same as graphing. When *graphing* a curve, precision is expected, so 'T'-charts or sketching points on the curve are normally required; however, when *sketching* graphs, you need only get an approximate graph of the curve.

Find

-intercept(s) and$x$ -intercept(s).$y$ Use known properties of known functions:

- Linear functions,
,$y=mx+b$ $ax+by+c=0$ - Quadratic functions,
$y=a{x}^{2}+bx+c$ - Polynomial functions,
$y={a}_{n}{x}^{n}+{a}_{n-1}{x}^{n-1}+\cdots +{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}$ - Root functions,
,$y=\sqrt{x}$ , for$y=\sqrt[n]{x}$ $n>1$ - Exponential functions,
,$y={a}^{x}$ $x={a}^{y}$ - Logarithmic functions,
,$y={\mathrm{log}}_{b}(x)$ $x={\mathrm{log}}_{b}(y)$ - Trigonometric functions

- Linear functions,
If necessary, exchange

and$x$ variables, sketch, then reflect graph over the$y$ line.$y=x$ Use your knowledge of translations of curves.

Plot point(s), only as a last resort.

Find the exact values of all **real**, if any solutions to the following equations:

${x}^{2}-5x=6$ ${x}^{3}=2$ $\sqrt{x+3}=7$ $\mathrm{ln}(x+1)=2$ $\mathrm{sin}(x)=\frac{-\sqrt{3}}{2}$ $\mathrm{sin}(x)=2$ ${\mathrm{tan}}^{-1}(x)=2$ ${e}^{x-7}=4$ $3{x}^{3}-{x}^{2}-3x+x=0$ ${x}^{2}=3$

**Mathematica**

`x1``(* Example from Practice 01-2: Solve x^3=2 *)`

2`(* NOTE: == is used for the equals sign in an equation *)`

3`Solve[x^3 == 2, x, Reals]`

4```
```

5`(* Example from Practice 01-4: Solve ln(x+1)=2 *)`

6`(* NOTE: == is used for the equals sign in an equation *)`

7`Solve[Log[x+1] == 2, x, Reals]`

8```
```

9`(* Example from Practice 01-5: Solve sin(x)=-sqrt{3}/2 *)`

10`(* NOTE: == is used for the equals sign in an equation *)`

11`Solve[{Sin[x] == -Sqrt[3]/2, 0 <= x < 2Pi}]`

12`(* or *)`

13`Reduce[{Sin[x] == -Sqrt[3]/2, 0 <= x < 2Pi}]`

**Warnings**:

- Be very
**careful**with the*syntax*.*Syntax*is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic*Mathematica*syntax is located at: http://www.jjw3.com/TECH_Common_Functions.pdf. - To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.
- When solve trigonometric equations, Mathematica output may include the symbol
which is the mathematical abbreviation for “or.”$\vee $

Find the exact values of all real intersection points:

;$y={x}^{2}$ $y=3+x$ ;$y=2-x$ $y={x}^{2}+1$ ;$y=\mathrm{sin}(x)$ $y=\frac{1}{2}$ ;$x={y}^{2}-2$ $x=y$ ;$x=\sqrt{y}$ $x=y$

**Mathematica**

`xxxxxxxxxx`

91`(* Example from Practice 02-4: Find intersection(s) of x=y^3-1 and x=y *)`

2`(* NOTE: == is used for the equals sign in an equation *)`

3`Solve[{x == y^2-2, x == y}, {x, y}, Reals]`

4```
```

5`(* Example from Practice 02-3: Find intersections of y=sin(x) and y=1/2 *)`

6`(* NOTE: == is used for the equals sign in an equation *)`

7`Solve[{Sin[x] == 1/2, 0 <= x < 2Pi}]`

8`(* or *)`

9`Reduce[{Sin[x] == 1/2, 0 <= x < 2Pi}]`

**Warnings**:

- Be very
**careful**with the*syntax*.*Syntax*is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic*Mathematica*syntax is located at: http://www.jjw3.com/TECH_Common_Functions.pdf. - To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.
- When solve for intersections of trigonometric equations, Mathematica output may include the symbol
which is the mathematical abbreviation for “or.”$\vee $

Sketch a graph of the following:

$y={x}^{3}+{x}^{2}-2x$ $y=\sqrt[3]{x}$ $y=\sqrt{x+5}$ $y=\mathrm{ln}(x+2)$ $y={e}^{-x}$ $y={\mathrm{tan}}^{-1}(x)$ $x={y}^{3}-2{y}^{2}$

**DESMOS**

**Warnings**:

- Be very
**careful**with the*syntax*.*Syntax*is the set of rules on how to write computer code. Every software program has its own unique syntax. For built-in*DESMOS*functions, click on the keyboard icon on the bottom-left of the page. *DESMOS*can graph more than one curve and can graph curves that are not functions.

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**Last Modified**: Monday, 6 September 2020 13:33 EDT