Question

Estimate the area under the graph of f(x)=1/x+4 over the interval [-1,2] using four approximating rectangles and right endpoints.

Answer 1

Answer:

  Rn ≈ 0.6345

  Ln ≈ 0.7595

Step-by-step explanation:

The interval from -1 to 2 has a width of (2 -(-1)) = 3. Dividing that into 4 equal intervals means each of those smaller intervals has width 3/4.

It can be useful to use a spreadsheet or graphing calculator to evaluate the function at all of the points that define these intervals:

  x = -1, -.25, 0.50, 1.25, 2

Of course, the spreadsheet can easily compute the sum of products for you.

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The approximation using right end-points will be the sum of products of the interval width (3/4) and the function value at the right end-points:

  Rn = (3/4)f(-0.25) +(3/4)f(0.50) +(3/4)f(1.25) +(3/4)f(2)

  Rn ≈ 0.6345

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The approximation using left end-points will be the sum of products of the interval width (3/4) and the function value at the left end-points:

  Ln = (3/4)f(-1) +(3/4)f(-0.25) +(3/4)f(0.50) +(3/4)f(1.25)

  Ln ≈ 0.7595

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It is usually convenient to factor out the interval width, so only one multiplication needs to be done: (interval width)(sum of function values).