Introduction to Integrals: Problems for "The Area as a Sum" 2

Problem : Consider the function f (x) = x² + 1 on the interval [0, 2]. Using four subdivisions, find the left-hand approximation, L₄, of the area under the curve of f on the interval indicated.

Δx		= = =
L₄		= f (0) + f () + f (1) + f ()
		= 1 + +2 +
		= =

Problem : For the same function, using four subdivisions, find the right-hand sum, R₄.

R₄		= f () + f (1) + f () + f (2)
		= +2 + + 5
		= =

Problem : For the same function, using four subdivisions, find the midpoint sum, M₄.

M₄		= f () + f () + f () + f ()
		= + + +
		= =

Notice that on the interval in question, f is a strictly increasing function. If f is increasing on an interval, then L_n < M_n < R_n. If f is decreasing on an interval, then R_n < M_n < L_n.

Problem : Find

f (x_k)Δx for f (x) = 2x on [0, 2]

To solve this problem, notice that the graph of f (x) is a line, and the area in question is in the shape of a right triangle with base 2 and height f (2) = 4. So, the limit of the right hand sum, which is the area under the curve, is

(2)(4) = 4

Problem : Find

f (x_k)Δx for f (x) =

on [0, 3]

To solve this problem, notice that the graph of f (x) is a semicircle of radius 3 centered at the origin. The interval [0, 3] contains a section of the curve that is equivalent to a quarter-circle. Thus, the area under the curve is equal to one-fourth the area of a circle with radius 3, or