Use the table of values to find lower and upper estimates of... Assume that f is a decreasing function.

Question

Use the table of values to find lower and upper estimates of

10

f(x)dx

0

.

Assume that f is a decreasing function.

lower estimate	?
upper estimate	?
x	0	2	4	6	8	10
f(x)	32	21	9	−4	−18	−37

Answer 1

Since f is decreasing, the minimum value of f on each subinterval occurs at the right endpoint of the subinterval and the maximum value of f on each subinterval occurs at the left endpoint.

lower sum = f(2)(2-0) + f(4)(4-2) + f(6)(6-4) + f(8)(8-6) + f(10)(10-8)

= 2(21 + 9 + (-4) + (-18) + (-37)) = -58

upper sum = f(0)(2-0) + f(2)(4-2) + f(4)(6-4) + f(6)(8-6) + f(8)(10-8)

= 2(32 + 21 + 9 + (-4) + (-18)) = 80

So, -58 ≤ ∫_{(0 to 10)} f(x)dx ≤ 80

Answer 2

Since f(x) is decreasing, the left Riemann sum would give an upper estimate and the right Riemann sum would give a lower estimate.

To calculate a Riemann sum, we are adding up areas of rectangles to estimate the area under the curve. The x-values in the table are the endpoints of the rectangles, so we have 5 rectangles, each with a width of 2. For the left Riemann sum, we choose the height of each rectangle to be the function value at the left endpoint of the rectangle. For the right Riemann sum, we choose the height of each rectangle to be the function value at the right endpoint of the rectangle.

Hope this helps!