3 Theorem If f is continuous on [a, b], or if f has only a finite number discontinuities, then f is integrable on [a, b]; that is, the definite integral exists. If f is integrable on [a, b], then the limit in Definition 2 exists and give value no matter how we choose the sample points x. To simplify the calcul integral we often take the sample points to be right endpoints. Then x = x,a nition of an integral simplifies as follows. 4 Theorem If f is integrable on [a, b], then Cod - |f(x) dx = lim E f(x) Ax i-1 where Ax = and Xi = a + i Ax EXAMPLE 1 Express lim (x + x, sin x,) Ax as an integral on the interval [0, 7]. SOLUTION Comparing the given limit with the limit in Theorem 4, we see t will be identical if we choose f(x) = x' + x sin x. We are given that a Therefore, by Theorem 4, we have %3D L² + "(x' (x+x sin x) dx lim (x + x, sin x,) Ax = 1-1 Lotar when we apply the definite integral to physical situations, it will be in did in Example 1. When Leibni eWe kaas that Si fde = Ax (atiax) flati Ax) %3D 下=L fC)こxtx b-a_of2) %3D धहे 2. (2+i-) lim (-2ti- lim T-1
Riemann Sum
Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence this method makes it simple. Usually, we take the help of different integration methods for this purpose. This is one of the major parts of integral calculus.
Riemann Integral
Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.
Use the form of the definition of the
I have included a picture of my work so far as well, and in the steps to solving this problem, can you especially please show me how we get from step 1 to step 2, where we square the (-2+(2i)/n)?
![3 Theorem If f is continuous on [a, b], or if f has only a finite number
discontinuities, then f is integrable on [a, b]; that is, the definite integral
exists.
If f is integrable on [a, b], then the limit in Definition 2 exists and give
value no matter how we choose the sample points x. To simplify the calcul
integral we often take the sample points to be right endpoints. Then x = x,a
nition of an integral simplifies as follows.
4 Theorem If f is integrable on [a, b], then
Cod -
|f(x) dx = lim E f(x) Ax
i-1
where
Ax =
and
Xi = a + i Ax
EXAMPLE 1 Express
lim (x + x, sin x,) Ax
as an integral on the interval [0, 7].
SOLUTION Comparing the given limit with the limit in Theorem 4, we see t
will be identical if we choose f(x) = x' + x sin x. We are given that a
Therefore, by Theorem 4, we have
%3D
L² +
"(x'
(x+x sin x) dx
lim (x + x, sin x,) Ax =
1-1
Lotar when we apply the definite integral to physical situations, it will be in
did in Example 1. When Leibni](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F88f794be-122d-44d8-bdbd-e59efaff39a0%2F3c925561-3cda-4620-b144-1d6670526443%2F9uh9tck.jpeg&w=3840&q=75)
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