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Find an approximation of the area of the region R under the graph of the function f on the interval [-1, 2]. Use n = 6 subintervals. Choose the representative points to be the left endpoints of the subintervals. f(x) = 7 - x2

Respuesta :

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What is the width of each interval?  To answer this, subtract -1 from 2.  Result:  3.  If n=6, then the width of each interval is 3/6, or 1/2.

If we're to use the left endpoints, then the x values in question are {-1, -1/2, 0, 1/2, 1, 3/2, 2}.

Evaluate the given function at these 6 x-values.  Answer set (to 2 decimal place accuracy):  {3,3.75, 4, 3.75, 3, 1.75).

Now calculate each of the 6 sub-areas under the curve.  To do this, multiply each of the 6 "heights" given immediately above by the subinterval width (1/2).  You will then have 6 subareas.  Find the sum of these 6 subareas.

Your result here is the approx. area under the curve of f(x)=7-x between x=-1 and x=2.
xero099

The approximation of the area of the region R under the graph of the function is 18.625 square units.

In this question we must calculate the approximate area ([tex]A[/tex]) below the curve by means of Riemann sums with left endpoints, whose expression is described below:

[tex]A = \Delta x \cdot \Sigma\limits^{n-1}_{i=0} f(a + i\cdot \Delta x)[/tex], for [tex]x \in [a,b][/tex] (1)

[tex]\Delta x = \frac{b-a}{n}[/tex] (2)

Where:

  • [tex]a[/tex] - Lower bound
  • [tex]b[/tex] - Upper bound
  • [tex]n[/tex] - Number of subintervals
  • [tex]i[/tex] - Index

If we know that [tex]a = -1[/tex], [tex]b = 2[/tex], [tex]n = 6[/tex] and [tex]f(x) = 7 - x^{2}[/tex], then the approximate area is:

[tex]\Delta x = \frac{2-(-1)}{6}[/tex]

[tex]\Delta x = 0.5[/tex]

[tex]A = 0.5\cdot [f(-1) + f(-0.5)+f(0)+f(0.5)+f(1)+f(1.5)][/tex]

[tex]A = 0.5\cdot (6+6.75+7+6.75+6+4.75)[/tex]

[tex]A = 18.625[/tex]

The approximation of the area of the region R under the graph of the function is 18.625 square units.

To learn more on Riemann sums, we kindly invite to check this verified question: https://brainly.com/question/21847158

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