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Chapter 4: Problem 67

Upper and Lower Sums In your own words and using appropriate figures, describe the methods of upper sums and lower sums in approximating the area of a region.

Short Answer

Expert verified
Upper sums and lower sums are mathematical techniques in calculus used to estimate the area under a curve over an interval. Upper sums use rectangles above the curve, influencing an overestimate, while lower sums use rectangles underneath the curve, causing an underestimate. As the number of rectangles or subintervals increase, the precision of these estimations improves, ultimately converging to the exact area under the curve given by the definite integral of the function.

Step by step solution

01

Defining Upper and Lower Sums

Upper sums and lower sums are techniques used in calculus to approximate the area under a curve. This is particularly useful when the actual area cannot be calculated directly. An upper sum is calculated by overestimating the region's area, typically using rectangles that lie above the curve within a prescribed interval. On the other hand, a lower sum underestimates the area by using rectangles that lie below the curve within the same interval.
02

Visual Representation of Upper and Lower Sums

To create visual representations of these terms, you typically sketch a graph of the function and denote an interval for which you are interested in calculating the area. You then partition the interval into subintervals. For upper sums, within each subinterval, you select a point with the maximum function value. You then construct a rectangle with base being the subinterval and height equal to this maximum function value. For the lower sum, within each subinterval, you select a point with the minimum function value and construct a rectangle with this minimum function value as the height.
03

Calculation of Upper and Lower Sums

Once all rectangles are drawn, calculate the areas of these rectangles and sum them up. The upper sum is the sum of the areas of the rectangles from the upper sum method, and the lower sum represents the sum of the areas of the rectangles from the lower sum method.
04

Refining the Approximation

As the number of rectangles (or the number of subintervals) increases, the approximation of the area under the curve using both the upper and lower sums becomes more accurate. In the limit, as the subintervals approach zero in width, the lower sum and upper sum converge to the value of the definite integral of the function over the interval, which represents the exact area under the curve.

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Most popular questions from this chapter

Finding Values In Exercises \(59-62,\) find possible values of \(a\) and \(b\) that make the statement true. If possible, use a graph to support your answer. (There may be more than one correct answer.) $$ \int_{-3}^{3} f(x) d x+\int_{3}^{6} f(x) d x-\int_{a}^{b} f(x) d x=\int_{-1}^{6} f(x) d x $$

Proof Prove that \(\int_{a}^{b} x^{2} d x=\frac{b^{3}-a^{3}}{3}\).

The velocity function, in feet per second, is given for a particle moving along a straight line. Find (a) the displacement and (b) the total distance that the particle travels over the given interval. \(v(t)=t^{3}-10 t^{2}+27 t-18, \quad 1 \leq t \leq 7\)

Approximating Area with the Midpoint Rule In Exercises \(61-64,\) use the Midpoint Rule with \(n=4\) to approximate the area of the region bounded by the graph of the function and the \(x\) -axis over the given interval. $$ f(x)=\tan x, \quad\left[0, \frac{\pi}{4}\right] $$

The velocity function, in feet per second, is given for a particle moving along a straight line. Find (a) the displacement and (b) the total distance that the particle travels over the given interval. \(v(t)=\cos t, \quad 0 \leq t \leq 3 \pi\)
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