Question

Find lower and upper bounds for the area between the z-axis and the graph of f(x) = √x+3

over the interval [-1, 1] by calculating left-endpoint and right-endpoint Riemann sums with 4
subintervals. The graphs of L4 and R4 are given below.

Answer 1

• L4 = 3.299
• R4 = 3.592

Explanation:

You want the left sum and the right sum of the four subinterval areas under the curve f(x) = √(x+3) on the interval [-1, 1].

Riemann sum

The Riemann sum is the sum of the subinterval areas. The area of each subinterval is the height of the rectangular area, multiplied by its width. Here, the interval width is (1 -(-1))/4 = 0.5.

The heights of the intervals of interest will be the function values f(-1 +0.5n) for f(x) = √(x+3) and n = 0 .. 3 for the left sum and 1 .. 4 for the right sum.

Values

The attached calculator display shows the function values for n=0 .. 4. The expression Total(Most( )) adds the first four function values; while the expression Total(Rest( )) adds the last four function values of these five. Multiplying by the interval width (1/2) gives the left- and right-Riemann sums, respectively.

• Lower Bound (L4) = 3.299
• Upper Bound (R4) = 3.592

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Additional comment

The actual integral value is about 3.44772.

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