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 1177659: 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: Use 10 rectangles to approximate the area under the curve.

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]