The mathematical expression shown in the image is an improper integral: \[ \int_{3}^{\infty} (x - 2)^{-7} \, dx \] Here's a step-by-step explanation for evaluating this integral: 1. **Set Up the Integral**: \[ \int_{3}^{\infty} (x - 2)^{-7} \, dx \] This integral is improper because the upper limit of integration is infinity. 2. **Rewrite the Integrand**: \[ (x - 2)^{-7} \] is the function we need to integrate. Another way to write this is \[ (x - 2)^{-7} = \frac{1}{(x - 2)^7} \]. 3. **Evaluating the Improper Integral**: First, convert the improper integral into a limit expression. We set up a limit where \( b \) approaches infinity: \[ \lim_{b \to \infty} \int_{3}^{b} (x - 2)^{-7} \, dx \] 4. **Integrate the Function**: To find the integral of \((x - 2)^{-7}\), we use the general power rule for integration, which states that \(\int (x - a)^n \, dx = \frac{(x - a)^{n+1}}{n+1} + C\), provided that \( n \neq -1 \). Applying this rule: \[ \int (x - 2)^{-7} \, dx = \frac{(x - 2)^{-7+1}}{-7 + 1} + C = \frac{(x - 2)^{-6}}{-6} + C = -\frac{1}{6} (x - 2)^{-6} + C \] 5. **Apply the Limits**: Now, substitute the limits into the integrated function: \[ \left[ -\frac{1}{6} (x - 2)^{-6} \right]_{3}^{b} = \lim_{b \to \infty} \left[ -\frac{1}{6} (b - 2)^{-6} - \left( -\frac{1}{6} (3 - 2)^{-6
The mathematical expression shown in the image is an improper integral: \[ \int_{3}^{\infty} (x - 2)^{-7} \, dx \] Here's a step-by-step explanation for evaluating this integral: 1. **Set Up the Integral**: \[ \int_{3}^{\infty} (x - 2)^{-7} \, dx \] This integral is improper because the upper limit of integration is infinity. 2. **Rewrite the Integrand**: \[ (x - 2)^{-7} \] is the function we need to integrate. Another way to write this is \[ (x - 2)^{-7} = \frac{1}{(x - 2)^7} \]. 3. **Evaluating the Improper Integral**: First, convert the improper integral into a limit expression. We set up a limit where \( b \) approaches infinity: \[ \lim_{b \to \infty} \int_{3}^{b} (x - 2)^{-7} \, dx \] 4. **Integrate the Function**: To find the integral of \((x - 2)^{-7}\), we use the general power rule for integration, which states that \(\int (x - a)^n \, dx = \frac{(x - a)^{n+1}}{n+1} + C\), provided that \( n \neq -1 \). Applying this rule: \[ \int (x - 2)^{-7} \, dx = \frac{(x - 2)^{-7+1}}{-7 + 1} + C = \frac{(x - 2)^{-6}}{-6} + C = -\frac{1}{6} (x - 2)^{-6} + C \] 5. **Apply the Limits**: Now, substitute the limits into the integrated function: \[ \left[ -\frac{1}{6} (x - 2)^{-6} \right]_{3}^{b} = \lim_{b \to \infty} \left[ -\frac{1}{6} (b - 2)^{-6} - \left( -\frac{1}{6} (3 - 2)^{-6
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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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.
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