(c) Find an upper estimate for the area bounded by the curve and the x-axis usin
ID: 2882298 • Letter: #
Question
(c) Find an upper estimate for the area bounded by the curve and the x-axis using n = 10 subintervals of equal length. Be sure to draw the 10 rectangles you used on the graph provided and show all of the terms you included in your computation. (d) Find a lower estimate for the area bounded by the curve and the r-axis using n 10 subintervals of equal length. Be sure to draw the 10 rectangles you used on the graph provided and show all of the terms you included in your computation. (e) What is different about the way you draw the rectangles when you are finding an upper estimate or lower estimate compared to how you draw the rectangles when using right or left endpoints to approximate the total area under a curve? Describe in 2-5 sentences. (f) Write, but do not evaluate, the integral(s) that would give the exact value of the area bounded by the curve and the x-axis.Explanation / Answer
f(x) = x^3 - 3x^2 on [-1 , 4]
the width of each subinterval would be = del(x)= (b-a)/n= (4-(-1))/10 = .5
the subintervals are : [-1 , -.5] , [-.5 , 0], [0, .5] , [.5 , 1], [1 , 1.5] , [1.5 , 2], [2 , 2.5] ,[2.5 , 3] ,[3 , 3.5] , [3.5, 4]
a>
from the above 10 subintervals take all the left endpoints and at those values find the value of f(x)
and those values would be the length of the 10 rectangles
so we have the length and the width , then we'll find the area of the 10 rectangles and finally add them up to get our approximation
f(-1)= -4
f(-.5) = -.875
f(0) = 0
f(.5) = -.625
f(1) = -2
f(1.5)= -3.375
f(2) = -4
f(2.5) = -3.125
f(3) = 0
f(3.5) = 6.125
=> area = .5(-4) + .5(-.875) + .5(0) + .5(-.625) + .5(-2) + .5(-3.375) + .5(-4) + .5(-3.125) + .5(0) + .5(6.125)
area bounded bu=y the curve = -5.9375 units^2
as more area is below the x axis so we get a negative area.
the negative sign indicates that more area is below the x axis
b>
width is del(x)=.5
from the above 10 subintervals take all the right endpoints and at those values find the value of f(x)
f(-.5) = -.875
f(0) = 0
f(.5) = -.625
f(1) = -2
f(1.5)= -3.375
f(2) = -4
f(2.5) = -3.125
f(3) = 0
f(3.5) = 6.125
f(4) = 16
=> area = .5(-.875) + .5(0) + .5(-.625) + .5(-2) + .5(-3.375) + .5(-4) + .5(-3.125) + .5(0) + .5(6.125) + .5(16)
area bounded by the curve = 4.0625 units^2
as more area is above the x axis so we get a positive area.
the positive sign indicates that more area is above the x axis
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