The area bounded between the line y=x+4 and the quadratic function y=(x^2)-2x.
Hint: Draw the region and find the intersection of the two graphs. Add and subtract areas until the appropriate area is found.
I found the intersection points as (-1,3) and (4,8).
I'm not sure what to do with this. I think the answer is 125/6, but not sure.
Please show me the steps on how to get this answer if it's correct.
Your intersection points are correct. Between those points, the y = x+4 curve is above the y = x^2 -2x curve.
The area between the curves is the integral of x + 4 dx from x = -1 to 4, MINUS the integral of x^2 -2x between the same two x values. That equals
x^2/2 + 4x @ x = 4 - (x^2/2 +4x) @ x = -1
MINUS
x^3/3 -x^2 @ x = 4 - (x^3/3 -x^2) @x = -1
Your intersections are correct.
the vertex of the parabola is at (1,-1)
The zeros of the parabola are at(0,0) and at (2,0)
Between x = 0 and x = 2, the parabola dips below the x axis
we want the height of the line between y = x+4 and y=x^2-2x
which is
x+4 -x^2+2x
or
-x^2 + 3x + 4
integrate that dx from -1 to +4
-x^3/3 + 3x^2/2 + 4x
at 4
-64/3 + 24 + 16 = (-64+72+48)/3
= 56/3
at -1
+1/3 +3/2 -4 = 2/6 + 9/6 - 24/6
= -13/6
so we want
56/3 -(-13/6) = 112/6+13/6 = 125/6
yep, agree 125/6
Yeah. Thanks for showing me the steps
To find the area bounded between the line y=x+4 and the quadratic function y=(x^2)-2x, you can follow these steps:
1. Begin by graphing the two functions on the same coordinate plane. This will allow you to visualize the region you need to find the area of.
2. Locate the points where the two graphs intersect. In this case, you correctly found that the intersection points are (-1,3) and (4,8).
3. Draw horizontal lines from the x-axis to the two intersection points. This will create two rectangles, one above the x-axis and one below it.
4. Calculate the area of the rectangles. To do this, find the length of each rectangle by subtracting the x-coordinates of the intersection points. The width of each rectangle is given by the y-values of the intersection points. Thus, the area of each rectangle is (length * width).
- The area of the rectangle above the x-axis is [(4-(-1)) * (8-4)] = 5 * 4 = 20.
- The area of the rectangle below the x-axis is [(4-(-1)) * (0-3)] = 5 * (-3) = -15. Note that the area is negative because it lies below the x-axis.
5. Combine the areas of the rectangles. Since the area below the x-axis is negative, we need to subtract it from the area above the x-axis to get the total area.
Total Area = (Area above x-axis) - (Area below x-axis)
= 20 - (-15)
= 20 + 15
= 35
So, the correct answer for the area bounded between the two graphs is 35.