The area of a definite integral

In summary, the problem involves computing an indefinite integral and interpreting it as areas. The function given is y = root(6 - 2x^2 + 4x) and the question asks for the area under the curve from x = -1 to x = 3. Relevant equations include the area of a circle, A = pi*r^2, and the area of a semi-ellipse, A = ab*pi. The attempt at a solution involves manipulating the function to find the location of the center and the radius of the semi-circle or semi-ellipse. Once the area of the semi-ellipse is found, it can be added to the area of a rectangle to get the total area under the curve. However, there is some
  • #1
Jack_B
5
0

Homework Statement



The problem requires the computation of an indefinite integral but says to "interpret as areas" which basically changes the question to the following;

What is the area under the curve of the function y = root( 6 - 2x2 + 4x) from x = -1 to x = 3

Homework Equations



Relevant equations would include the area of a circle, A=pi*r2

The Attempt at a Solution



To begin with, I am not sure if the function is just a semi circle or also extends beyond just half of a circle. That being said, with the assumption that the function is a semi circle, the diameter is 4 (3 - (-1)), thus the radius is 2, making the total area (1/2)pi*22. However, with a graphing calculator, I can see that the area is 8.89.

Therefore I am led to believe that the area under the curve is a semicircle + a rectangular area. However, I am unsure as to how I can find the radius of the semicircle and the height of the rectangle.
 
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  • #2
try completing the square
[tex]6+4x-2x^{2}=2(3+2x-x^{2})=2(4-(x-1)^{2})[/tex]
 
  • #3
Unfortunately, integration is not allowed. The question must be completed by interpreting the area under the function as a sum of areas (a semicircle and perhaps a rectangle) but I have no idea how to, using the given function, find the semi circle's radius to be able to find its area. Unless completing the square puts the equation in a form where the semicircle's radius is more apparent, I'm not sure how it would help.
 
  • #4
But it does; it gives you the location of the center and from there you can figure out the radius.
 
  • #5
zcd said:
But it does; it gives you the location of the center and from there you can figure out the radius.



Really?

This is the first time in my life I've seen a semicircle function. With that manipulation, what is the location of the centre?

Thank you so much, I've been stumped on this one for about an hour.
 
  • #6
[tex]y=\sqrt{2(4-(x-1)^{2})}\implies y^{2}=2(4-(x-1)^{2})[/tex] with some minor caveats in the range. [tex]y^{2}+2(x-1)^{2}=8[/tex] is an equation of?
 
  • #7
It seems like the equation of a circle, the only equation of a circle I've ever seen is y^2 + x^2 = 9 which has a radius of 3 and center of (0, 0). But I'm still not quite sure how to obtain the location of the center of the circle from the newly manipulated one you gave me. It seems as the y coordinate of the center would still be 0, but I don't know about the x coordinate of the origin.

After some googling, I think that the y coordinate is 0 and the x coordinate is 1, but I'm not sure how the factor of 2 affects it.
 
  • #8
Ah, apparently it's an equation of a semi-ellipse. Any idea as to how I can find its area? Either if a formula for it exists or if I must split it up into common geometric shapes and take their sum?
 
Last edited:
  • #9
[tex]A=ab\pi[/tex] where a and b are the lengths of semi axes
 

Related to The area of a definite integral

What is the definition of "the area of a definite integral"?

The area of a definite integral is a numerical value that represents the area under a curve between two specific points on the x-axis. It is calculated by finding the value of a definite integral, which is the limit of a Riemann sum.

How is the area of a definite integral calculated?

The area of a definite integral is calculated by taking the integral of a function between two specified limits. This involves finding the antiderivative of the function and evaluating it at the upper and lower limits, then taking the difference between the two values.

What is the significance of the area under a curve in a definite integral?

The area under a curve in a definite integral represents the total amount of change in the function over a specific interval. It can also be interpreted as the net accumulation of the function over that interval.

Can the area of a definite integral be negative?

Yes, the area of a definite integral can be negative if the function being integrated has negative values between the specified limits. This indicates that the function is decreasing over that interval.

What is the relationship between the area of a definite integral and the derivative of a function?

The area of a definite integral and the derivative of a function are closely related. The derivative of a function represents the rate of change of the function, while the area under the curve of the derivative represents the total change in the original function over a specific interval.

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