Calc 2 recitation 2 PDF

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Jose Ramirez Math 152 Section 26 Recitation Workshop 1

Question 5: Let R be the region bounded by the parabola y = x - x 2 and the x-axis. Find he equation of the line through the origin that divides R into two subregions of equal area.

To first approach this problem, we’ll have to know what the bounded region is and what the total area within that bound is. To know what the bounded region is in the parabola of y=x- x 2 we’ll have to use some algebra to find where the function crosses the x-axis. This will let us know between what bounds we can find an area underneath the curve but above the x-axis. Simply graphing the function could help you find between what bound(s) this area is located as well. To find where the function is above the x-axis we need to find the zeros of the function, you have set the equation equal to zero: y=x-

x

x(1-x) = 0

2

0 = x-

x

2

⁃ Factor out an x

-› x=1 x=0 ⁃ This yields to the function crossing at x=0 and x=1. From this, we’re able to estimate what the graph will look like.

Sine - x 2 is negative, this means that the function will be a concave down parabola meaning that the vertex will be at the highest point on the graph. To find the vertex of the function, you −b to give you the x-coordinate of the vertex, you then plug the x-value use the expression 2a 1 1 , ). The graph looks like that you got into your function and get the vertex coordinates of ( 2 4 this:

Now we know where our area is located. It’s located between x=0 and x=1. From this we can simply integrate the function between those bounds. We set up our integral to find the total area. Since the question tells us that the area is cut in half, we need to find the total area first and then divide it by two t help determine the equation of the line. This is our area: 1

∫ x − x 2 dx= 0

3

x2 x − 2 3

=

}

1 1 − 2 3

1 0

=

1 6

This is our total area. Now we have to divide it by two in order to know what half of the area is. 1 1 by 2 give us an area of . This is how much each area is going to be. Next, to help Dividing 12 6 find what the line will be, we have to know at what point the line will intersect with the curve. We know that the line has to go through the origin so we can use the Fundamental Theorem of calculus to help us determine where on the curve and the line intersect. It intersects at some point xo .

As seen on the graph, the line will have to cross the origin and the point on the curve. By using The Fundamental theorem of calculus, we’re able to find the slope of the line. With the 1 is our integral will be information we found, we know that the area of both regions is 12 equal to this...