Finding x of line bisecting parabola

In summary, the problem involves finding the x-coordinate of point A on a parabola, given that the line OA divides the shaded area into two equal parts. The area of the parabola is 4 units^2 and its equation is -3(x^2-2x). This can be solved using calculus or by completing the square.
  • #1
TheFallen018
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Hey, I have this question I've been trying to figure out in an integration textbook. The part of the question that I'm having trouble understanding is basically this.

With the parabola below, find the x coordinate of A, if the line OA divides the shaded area into two equal parts.
View attachment 7519

The area of the parabola is 4 units^2.

The equation of the function is -3(x^2-2x))Would someone be so kind as to help me out with this one. I'm even quite sure how to approach this. Thanks.
 

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  • #2
The area under a curve, y= f(x), and above the x-axis between x= a and x= b is given by $\int_a^b f(x) dx$. Whoever gave you this problem expected you to know that! Taking $x_0$ to be the x-coordinate of point A, the area of the region to the left is $-3 \int_0^{x_0} x^2- 2x dx$. The area of the region to the right is $-3\int_{x_0}^2 x^2- 2x dx$. Do those two integrals, set them equal and solve for $x_0$.

(Now that I have said all that, I notice this can be done without any Calculus at all! A parabola is symmetric about its axis. The x-coordinate that divides the parabola into two equal parts is the x-coordinate of the vertex. you can get that by "completing the square". Calculate it both ways and see if you get the same answer.)
 
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  • #3
TheFallen018 said:
Hey, I have this question I've been trying to figure out in an integration textbook. The part of the question that I'm having trouble understanding is basically this.

With the parabola below, find the x coordinate of A, if the line OA divides the shaded area into two equal parts.The area of the parabola is 4 units^2.

The equation of the function is -3(x^2-2x))Would someone be so kind as to help me out with this one. I'm even quite sure how to approach this. Thanks.

You might find this thread helpful:

http://mathhelpboards.com/questions-other-sites-52/phyllis-question-yahoo-answers-regarding-finding-line-divides-area-into-equal-parts-7197.html
 
  • #4
HallsofIvy said:
Taking $x_0$ to be the x-coordinate of point A, the area of the region to the left is $-3 \int_0^{x_0} x^2- 2x dx$. The area of the region to the right is $-3\int_{x_0}^2 x^2- 2x dx$. Do those two integrals, set them equal and solve for $x_0$
The problem states that $\overline{OA}$ divides the parabola into two equal parts, not that $x=A$ divides the parabola into two equal parts.
 
  • #5
Yes, I misread the problem. Thanks. Taking A to have coordinates $(x_0, 6x_0- 3x_0^2)$, the line OA has equation [tex]y= (6- 3x_0)x[/tex]. The area under the parabola and above that line is given by $\int_0^{x_0} 6x- 3x^2- (6- 3x_0)x dx= \int_0^{x_0} 3x_0x- 3x^2 dx$. Since you already know that the area of the parabola is 4, set that integral equal to 2 and solve for $x_0$.
 

FAQ: Finding x of line bisecting parabola

1. What is the equation for a line bisecting a parabola?

The equation for a line bisecting a parabola is y = mx + b, where m is the slope of the line and b is the y-intercept.

2. How do I find the slope of the line bisecting a parabola?

The slope of the line bisecting a parabola can be found by taking the derivative of the parabola's equation and setting it equal to the slope equation, y = mx + b. The resulting value for m will be the slope of the line.

3. Can a line bisect a parabola at more than one point?

Yes, a line can bisect a parabola at more than one point. This occurs when the parabola is symmetric and the line passes through the vertex of the parabola.

4. How do I find the y-intercept of the line bisecting a parabola?

The y-intercept of the line bisecting a parabola can be found by plugging in any x-value from the line's equation into the parabola's equation. The resulting y-value will be the y-intercept.

5. Is there a specific method for finding the line that bisects a parabola?

Yes, there is a specific method for finding the line that bisects a parabola. This method involves using the vertex of the parabola and the equation for the axis of symmetry to determine the slope and y-intercept of the line.

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