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 Improper
the bounds are 3 to infinity.
![The integral provided is:
\[ \int_{3}^{\infty} (x - 2)^{-7} \, dx \]
This is a definite integral with the lower limit of integration at \( x = 3 \) and the upper limit extending to infinity. The integrand is \( (x - 2)^{-7} \), which is the function to be integrated with respect to \( x \). This integral represents the area under the curve of \( (x - 2)^{-7} \) from \( x = 3 \) to \( x \) approaching infinity.
On an educational website, this integral might be used in the context of improper integrals, which deal with limits that approach infinity or integrals over unbounded intervals. The solution of such integrals often involves finding convergence or divergence of the integral.
To solve, one approach would be to use substitution or recognizing it as a standard form integral. Note that the power of the integrand is negative, which might affect the convergence based on the properties of improper integrals.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9a36eedc-1311-4ac7-908a-8a22bc0a900c%2Fc6317d97-08de-43e6-9638-896cdb366698%2Fpvwqgxr_processed.png&w=3840&q=75)
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