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Answer:
a) integral = 24.72
b) |Error| ≤ 0.4267
Step-by-step explanation:
a)
The integral:
can be approximated with the midpoint rule, as follows:
6.7*(0.8 - 0.0) + 8.9*(1.6 - 0.8) + 6.9*(2.4-1.6) + 8.4*(3.2 - 2.4) = 24.72
(that is, all the intervals are 0.8 units length and f(x) is evaluated in the midpoint of the interval)
b)The error bound for the midpoint rule with <em>n</em> points is:
|Error| ≤ K*(b - a)^3/(24*n^2)
where <em>b</em> and are the limits of integration of the integral and K = max |f''(x)|
Given that -5 ≤ f''(x) ≤ 1, then K = 5. Replacing into the equation:
|Error| ≤ 5*(3.2 - 0)^3/(24*4^2) = 0.4267
Step-by-step explanation:
OPQ originally forms a sector, the formula for sector is
where x is the degrees of rotation between the two radii.
We know three sides length and is trying to find an angle between the radii so we can use law of cosines which states that
This isn't the standard formula, it's for this problem
So we found our angle of rotation, which is 106.26
Now, we do the sector formula.
is the area of sector.
Now let find the area of triangle, we can use Heron formula,
The area of a triangle is
where s is the semi-perimeter.
To find s, add all the side lengths up, 10,10,16 and divide it by 2.
Which is
So our area of the triangle is 48. Now, to find the shaded area subtract the main area,( the sector of the circle) by the area of the triangle so we get
Which is an approximate or
44.73