Given the function 6x+3y=18, find the area under the curve from x=0 to x=3 using 3 strips of equal length using the right end points.
I was working on this and so I made the equation equal to Y=mx+b form so it ends up being Y=-2x+6, and now I am completely confused.. Please help!
Your 3 strips would be 3 trapezoids.
The confusing part is the fact that the last "trapezoid" is actually just a triangle
area of trapezoids
= (1/2)(1)(6+4) + (1/2)(1)(4+2) + (1/2)(1)(2+0)
= 5 + 3 + 1
= 9
( your area consists of a triangle with a base of 3 and a height of 6
area = (1/2)base x height
= (1/2)(3)(6) = 9 , confirming our answer above )
I had wrote 9 as my answer for this question when I had tried it myself last week and my teacher gave it back to me yesterday saying it was wrong, which is Y I was also confused and decided to post the question on here , I'm very confused ..
To find the area under the curve of the function 6x + 3y = 18, you need to first rearrange the equation in terms of y.
Let's start from the original equation:
6x + 3y = 18
To solve for y, isolate it by moving the x term to the other side:
3y = 18 - 6x
Divide both sides of the equation by 3 to get y alone:
y = (18 - 6x) / 3
Simplifying further, we have:
y = 6 - 2x
Now that we have the equation in the form y = mx + b, with m = -2 and b = 6, we can proceed with finding the area under the curve.
To approximate the area under the curve from x = 0 to x = 3 using 3 equal strips and the right end points, you can use the right Riemann sum method.
The width of each strip can be calculated by dividing the total width (x = 3 - x = 0 = 3) by the number of strips (3).
Width of each strip = (3 - 0) / 3 = 1
Now, we can proceed to calculate the area of each strip by multiplying the width by the height. In this case, the height is the value of the function at the right endpoint of each strip.
Let's divide the interval [0, 3] into 3 equal strips (with width = 1) and find the height at each right endpoint.
For the first strip (from x = 0 to x = 1):
Substitute x = 1 into the equation y = 6 - 2x:
y1 = 6 - 2(1) = 4
For the second strip (from x = 1 to x = 2):
Substitute x = 2 into the equation y = 6 - 2x:
y2 = 6 - 2(2) = 2
For the third strip (from x = 2 to x = 3):
Substitute x = 3 into the equation y = 6 - 2x:
y3 = 6 - 2(3) = 0
Now, we can calculate the area of each strip:
Area of the first strip = width * height = 1 * 4 = 4
Area of the second strip = width * height = 1 * 2 = 2
Area of the third strip = width * height = 1 * 0 = 0
Finally, to find the total area under the curve, add up the areas of all the strips:
Total area = Area of the first strip + Area of the second strip + Area of the third strip
Total area = 4 + 2 + 0 = 6
Therefore, the area under the curve from x = 0 to x = 3, using 3 strips of equal length with right end points, is 6 units squared.