# Area of the bounded regions between a straight line and a polynomial

• MHB
• anemone
In summary, the area of the bounded regions between a straight line and a polynomial is the total space enclosed by the straight line and the curve created by the polynomial. It can be calculated by finding the definite integral of the polynomial function within the given bounds of the straight line. This concept is useful in determining volume and can be applied in various fields such as engineering, physics, and economics. It cannot have a negative value as it represents a physical space.
anemone
Gold Member
MHB
POTW Director
Let $P$ be a real polynomial of degree five. Assume that the graph of $P$ has three inflection points lying on a straight line. Calculate the ratios of the areas of the bounded regions between this line and the graph of the polynomial $P$.

Let $P = f(x)$ be the polynomial.

Changing coordinates if necessary, we can assume that the middle one of the three inflection points is at the origin. Let the other two inflection points be at $x=-a$ and $x=b$, where $a,b>0$.

Then $f''(x) = kx(x+a)(x-b) = k(x^3 + (a-b)x^2 - abx)$. The value of the constant $k$ does not affect the calculations except to cause clutter, so I will assume that $k=1$. Then $f(x) = \frac1{20}x^5 + \frac1{12}(a-b)x^4 - \frac1{6}abx^3 + cx$ for some constant $c$. (The constant term in $f(x)$ is zero because the curve passes through the point of inflection at the origin.)

Next, $f(-a) = -\frac1{20}a^5 + \frac1{12}(a-b)a^4 + \frac1{6}a^4b - ca = \frac1{30}a^5 + \frac1{12}a^4b - ac$, and similarly $f(b) = -\frac1{30}b^5 - \frac1{12}a^4b + bc$. But the points $(-a,f(-a))$ and $(b,f(b))$ lie on a straight line through the origin. Therefore $\dfrac{f(-a)}{-a} = \dfrac{f(b)}b$, so that $\frac1{30}a^5 + \frac1{12}a^4b = \frac1{30}b^5 + \frac1{12}a^4b,$ which simplifies to $(b^2 - a^2)(12a^2 + 30ab + 12b^2) = 0.$ Since the second term in that product is positive, it follows that $b^2 - a^2=0$ and so $b=a$. Therefore $f(x) = \frac1{20}x^5 - \frac1{6}a^2x^3 + cx$.

[TIKZ][scale=3]\draw [help lines, ->] (-1.75,0) -- (1.75,0) ;
\draw [help lines, ->] (0,-1) -- (0,1) ;
\draw [help lines] (-1,0) -- (-1,0.5) ;
\draw [help lines] (1,-0.5) -- (1,0) ;
\draw[ domain=-1.75:1.75, samples=100] plot (\x,0.6*\x^5 - 2*\x^3 + \x);
\draw (-1.75, 0.7) -- (1.75,-0.7) ;
\draw (-1,-0.1) node {$-a$} ;
\draw (1,0.1) node {$a$} ;
\draw (-1.3,0.7) node {$\color{red}A$} ;
\draw (-0.5,0) node {$\color{red}B$} ;
\draw (0.5,0) node {$\color{red}C$} ;
\draw (1.3,-0.7) node {$\color{red}D$} ;[/TIKZ]

The difference between the quintic polynomial and the straight line is $\frac1{60}x(3x^4 - 10a^2x^2 + 7a^4) = \frac1{60}x(x^2-a^2)(3x^2 - 7a^2)$. The points of intersection are at $x = \pm a$ and $x = \pm\sqrt{\frac73}a$. Integrating over the appropriate intervals, I found the ratios of the areas $A:B:C:{D}$ to be $32:81:81:32$. But my arithmetic is unreliable, so please check those numbers.

## 1. What is the formula for finding the area of the bounded regions between a straight line and a polynomial?

The formula for finding the area of the bounded regions between a straight line and a polynomial is the integral of the polynomial function minus the integral of the straight line function.

## 2. How do you graph the bounded regions between a straight line and a polynomial?

To graph the bounded regions, plot the straight line and polynomial functions on the same coordinate plane. Then, shade the area between the two curves to represent the bounded regions.

## 3. Can the area of the bounded regions between a straight line and a polynomial be negative?

Yes, the area of the bounded regions can be negative if the polynomial function is above the straight line function for some intervals and below it for others. This results in the integral of the polynomial being greater than the integral of the straight line, resulting in a negative value.

## 4. How does the degree of the polynomial affect the area of the bounded regions?

The degree of the polynomial affects the complexity of the function and the number of bounded regions. A higher degree polynomial may have more bounded regions, resulting in a larger total area.

## 5. Can the area of the bounded regions between a straight line and a polynomial be approximated?

Yes, the area can be approximated by using numerical methods such as Riemann sums or the trapezoidal rule. These methods divide the bounded region into smaller rectangles or trapezoids and calculate the approximate area by adding the areas of these shapes.

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