calculus area question

Question

Estimate the area under the graph of the function f(x)=x+2−−−−−√ from x=−1 to x=3 using a Riemann sum with n=10 subintervals and midpoints. Round your answer to four decimal places. area =

Answer 1

It's a little hard to decipher the function in your question but I'll assume you intended the following:

f(x) = √(x + 2)

Since we are integrating between -1 and 3, the width of each rectangle (Δx) would be:

Δx = (3 - (-1))/10 = 0.4 with the midpoints being, of course in the middle of that. The breakdown would look like this then:

Start | End | Midpt | f(midpt)

-1.0 | -0.6 | -0.8 | f(-0.8) = 1.095

-0.6 | -0.2 | -0.4 | f(-0.4) = 1.265

-0.2 | 0.2 | 0.0 | f(0.0) = 1.414

0.2 | 0.6 | 0.4 | f(0.4) = 1.549

0.6 | 1.0 | 0.8 | f(0.8) = 1.673

1.0 | 1.4 | 1.2 | f(1.2) = 1.789

1.4 | 1.8 | 1.6 | f(1.6) = 1.897

1.8 | 2.2 | 2.0 | f(2.0) = 2.000

2.2 | 2.6 | 2.4 | f(2.4) = 2.098

2.6 | 3.0 | 2.8 | f(2.8) = 2.191

Each rectangle is 0.4 wide and f(midpoint) tall so the area of the first rectangle is (0.4)(1.095) = 0.438 square units. Now calculate each area and add them all up:

Total area = (0.4)(1.095) + (0.4)(1.265) + (0.4)(1.414) + (0.4)(1.549) + (0.4)(1.673) + (0.4)(1.789) + (0.4)(1.879) + (0.4)(2.000) + (0.4)(2.098) + (0.4)(2.191)

Answer 2

I'm assuming the function f(x) in question is √(x + 2)

Interval [a, b] is [-1, 3] as given in the problem.

n = 10

The width of each subinterval, Δx is (b - a) / n = 0.4.

The midpoint of each subinterval is x_i* = a + (i + 0.5)Δx for i = 0, 1, 2,..., 9

x₀* = -1 + 0.5 × 0.4 = -0.8

x₁* = -1 + 1.5 × 0.4 = -0.4

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x₉* = -1 + 9.5 × 0.4 = 2.8

Then, evaluate the function at each midpoint:

f(x₀*) = √1.2

f(x₁*) = √1.6

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f(x₉*) = √4.8

Then, calculate the Midpoint Riemann sum:

Area = [∑(i = 0, 9) of f(x_i*)] × Δx

Area = [√1.2 + √1.6 + ... + √4.8] × 0.4

You should end up with a value of 6.7888, which is the estimated area under the graph.

Hope this was helpful.