Kenneth S. answered • 09/04/16

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4) Given that 7 ≤ f(x) ≤ 8 for -5 ≤ x ≤ 5, estimate the value of ∫ from 5 to -5 of f(x) dx.

*This just means that the y values of the function are all bedtween 7 and 8, over the interval [-5,5]. If we use the lowest possible y, then the def. integral is the area of a rectangle having base length 10, height 7, or minimal area under curve is 70. Using max. possible y of 8, then the area maximum would be 10(8).*

The answer has two blanks to put the answer in...

**70**≤ ∫ from 5 to -5 of f(x) dx ≤

**80**.

2) A table of values of a decreasing function f is shown. Use the table to find a lower and an upper estimate for the definite integral from 8 to -10 of f(x) dx. There is a chart & I will do my best to relay it.

x= -10 -7 -4 -1 2 5 8

f(x)= 9 5 2 -3 -4 -8 -15

Lower Estimate?

Upper Estimate?

*Using similar reasoning that I employed 4) above, I will do the estimates for from -10 to 8, i.e. normal L to R order of areas of rectangles as an area estimate. Note that the rectangles will all have Δx = 3. In this first estimate, I'll use left endpoints:*

*First Est. = 9(3) + 5(3) + 2(3) + -3(3) + -4(3) + -8(3) ...(you do the arithmetic!)*

*Second Est. using right endpoints: = 5(3) + 2(3) + -3(3) + -4(3) + -8(3) + -15(3) ...*

*Now, a word of caution: The question wanted the def. integration to proceed from R to L, so the above answers must have their signs changed! The lower final estimate would be the most negative of these revised two numbers.*