Area Between 2 Curves Homework: Calculating the Area between Two Tangent Curves

In summary: The shaded area in the graph.In summary, the problem is not well-posed. There is an infinite area between the line and the parabola, and the region between the parabola and the line is in two parts -- the roughly triangular piece to the right of the point where the two figures intersect, and the part to the left of the intersection point.
  • #1
squenshl
479
4

Homework Statement


Calculate the area between ##y = -x^2+3x+10## and ##y = -x+14##. Note that ##y = -x+14## is the tangent to the curve ##y = -x^2+3x+10## at the point ##(2,12)##.

Homework Equations

The Attempt at a Solution


Is it as simple as calculating ##\int_{5}^{14} -x+14+x^2-3x-10 = \int_{5}^{14} x^2-4x-10##?
 
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  • #2
squenshl said:

Homework Statement


Calculate the area between ##y = -x^2+3x+10## and ##y = -x+14##. Note that ##y = -x+14## is the tangent to the curve ##y = -x^2+3x+10## at the point ##(2,12)##.

Homework Equations

The Attempt at a Solution


Is it as simple as calculating ##\int_{5}^{14} -x+14+x^2-3x-10 = \int_{5}^{14} x^2-4x-10##?
Where did the 5 come from?

Anyway, your problem is not well-posed, as posted here. Is there some additional information that you left out? The area of the region between the line and the parabola is infinitely large. Did you sketch a graph?
 
  • #3
squenshl said:

Homework Statement


Calculate the area between ##y = -x^2+3x+10## and ##y = -x+14##. Note that ##y = -x+14## is the tangent to the curve ##y = -x^2+3x+10## at the point ##(2,12)##.

Homework Equations

The Attempt at a Solution


Is it as simple as calculating ##\int_{5}^{14} -x+14+x^2-3x-10 = \int_{5}^{14} x^2-4x-10##?

it depends on what you want out of this problem.

your problem statement didn't say that the area is to be calculated over ##[5, 14]##, and you are missing a dx in those integrals, but otherwise it looks basically ok.

two concerns:

the problem statement asks for

##\int_{a}^{b} \big \vert (-x+14) - (-x^2+3x+10) \big \vert dx##

how do you know that

##\big \vert (-x+14) - (-x^2+3x+10) \big \vert = (-x+14) - (-x^2+3x+10) ##

?

Put differently, how do you know you can just get rid of those absolute value signs? In general this would require a justification. I would always start by graphing it / drawing a picture, but some symbolic / mathematical justification is still needed.

you may also want to confirm ##14 - 10 \neq -10##
 
  • #4
Sorry about the confusion.
The area is between the line and the parabola and above the x axis. My bounds were ##5 \leq x \leq 14## because that's where the line and curve cut the x-axis after drawing it.
 
  • #5
squenshl said:
Sorry about the confusion.
The area is between the line and the parabola and above the x axis.
That's still an infinite area, unless there are some other bounds. The region between the parabola and the line, and above the x-axis, is in two parts -- the roughly triangular piece to the right of the point where the two figures intersect, and the part to the left of the intersection point. The portion on the left is infinite in area.
 
  • #6
Mark44 said:
That's still an infinite area, unless there are some other bounds. The region between the parabola and the line, and above the x-axis, is in two parts -- the roughly triangular piece to the right of the point where the two figures intersect, and the part to the left of the intersection point. The portion on the left is infinite in area.
 

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  • #7
your problem statement still is underspecified. Where did the integration bounds of [5,14] come from? For instance, why not also integrate from ##(-\infty, -2)## and why not [-2,2]?
 
  • #8
StoneTemplePython said:
your problem statement still is underspecified. Where did the integration bounds of [5,14] come from? For instance, why not also integrate from ##(-\infty, -2)## and why not [-2,2]?
Thanks for the help.
I just calculated ##\int_{2}^{15} -x+14 \; dx## and ##\int_{2}^{5} -x^2+3x-10 \; dx## then took the difference to get ##\frac{329}{6}##.
 
  • #9
squenshl said:
I just calculated ##\int_{2}^{15} -x+14 \; dx## and ##\int_{2}^{5} -x^2+3x-10 \; dx## then took the difference to get ##\frac{329}{6}##.
There is also the area to the left of the parabola and below the line.
 
  • #10
Mark44 said:
There is also the area to the left of the parabola and below the line.
Not according to the graph above.
 
  • #11
squenshl said:
Not according to the graph above.
But the graph doesn't agree with your problem statement in post #1.
squenshl said:
Calculate the area between ##y = -x^2+3x+10## and ##y = -x+14##.
The area between the line and the parabola includes the portion to the left of the parabola.
The graph implies additional restrictions that aren't given in the problem statement -- i.e., that ##x \ge 2## and ##y \ge 0##.

So which one is the problem you're working? The verbal description or the graph? They aren't the same.
 
  • #12
StoneTemplePython said:
Where did the integration bounds of [5,14] come from?

squenshl said:
I just calculated ##\int_{2}^{15} -x+14 \; dx## and ##\int_{2}^{5} -x^2+3x-10 \; dx## then took the difference to get ##\frac{329}{6}##.
The 15 in the first integral will give you a wrong answer. Was that a typo?
 
  • #13
Sorry guys.
The question was to find the shared area in the graph & yes that was a typo.
Cheers.
 
  • #14
squenshl said:
The question was to find the shared area in the graph
Do you mean "shaded" area?
 

Related to Area Between 2 Curves Homework: Calculating the Area between Two Tangent Curves

1. What is the formula for finding the area between two curves?

The formula for finding the area between two curves is ∫ab(f(x) - g(x))dx, where a and b are the points of intersection between the two curves and f(x) and g(x) are the equations of the curves.

2. How do you determine which curve is on top when finding the area between two curves?

To determine which curve is on top, you can graph the two curves and visually identify which one is above the other at each point of intersection. Alternatively, you can solve for the points of intersection and plug in values to see which curve gives a larger output.

3. Can there be negative area between two curves?

Yes, there can be negative area between two curves. This occurs when the lower curve is above the upper curve at certain points of intersection, resulting in negative values when calculating the area using the formula.

4. What is the significance of finding the area between two curves?

Finding the area between two curves can be useful in many applications, such as in physics and engineering, where it can represent the displacement of an object or the work done by a force. It can also be used to find the volume of a solid of revolution or to determine the probability of an event in statistics.

5. Can the area between two curves be infinite?

Yes, the area between two curves can be infinite. This occurs when one or both of the curves extend to infinity or when the curves have infinitely many points of intersection. In these cases, the integral used to calculate the area will also be infinite.

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