SOLUTION: Background info: In this task, you will practice finding the area under a nonlinear function by using rectangles. You will use graphing skills in addition to the knowledge gathered

Algebra ->  Rectangles -> SOLUTION: Background info: In this task, you will practice finding the area under a nonlinear function by using rectangles. You will use graphing skills in addition to the knowledge gathered      Log On


   

Question 1177660: Background info: In this task, you will practice finding the area under a nonlinear function by using rectangles. You will use graphing skills in addition to the knowledge gathered in this unit. Sketch the graph of the function y = 20x − x2, and approximate the area under the curve in the interval [0, 20] by dividing the area into the given numbers of rectangles.
question: Calculate the area under the curve using rectangles as their number becomes arbitrarily large (tends to infinity). You do not need to sketch the rectangles.



Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

Sketch the graph of the function y+=+20x-x%5E2

graph%28+600%2C+600%2C+-10%2C+35%2C+-10%2C+110%2C+20x-x%5E2%29+

divide the area in 10 rectangles


In order to figure the width of each rectangle we can use the following formula:
Δ x=%28b-a%29%2Fn
in this case a=0,+b=20 and n=10 so we get:
Δ x=%2820-0%29%2F10=2
so each rectangle must have a width of 2+units
We can now calculate the height of each rectangle. So we figure the y-value of each corner of the rectangles. We get the following heights:
h%5B1%5D=36 ..........x=2+units->y+=+20%2A2-2%5E2=36
h%5B2%5D=64..............x=4+units->y+=+20%2A4-4%5E2=64
h%5B3%5D=84.............x=6+units->y+=+20%2A6-6%5E2=84...and so on
h%5B4%5D=+96
h%5B5%5D=100
h%5B6%5D=96
h%5B7%5D=84
h%5B8%5D=64
h%5B9%5D=36
h%5B10%5D=0
so now we can use the following formula to find the area under the graph. Basically what the formula does is add the areas of the rectangles:

A=2%2836%2B64%2B84%2B96%2B100%2B96%2B84%2B64%2B36%2B0%29
A=1320 -> approximate the area under the curve in the interval [0, 20]