1) Given the curve y = x² over the interval [1,3], (a) Find the exact value for the area under the curve using the Riemann sums definition of an integral using right endpoints of each subinterval as sample points. Show details carefully. (b) Find the area exactly using the FTC part 2 and the antiderivative. (i.e. integrate directly) *
1) Given the curve y = x² over the interval [1,3], (a) Find the exact value for the area under the curve using the Riemann sums definition of an integral using right endpoints of each subinterval as sample points. Show details carefully. (b) Find the area exactly using the FTC part 2 and the antiderivative. (i.e. integrate directly) *
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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![**Calculating the Area under the Curve using Different Methods**
**1. Given the curve \( y = x^2 \) over the interval \([1,3]\):**
**(a) Find the exact value for the area under the curve using the Riemann sums definition of an integral using right endpoints of each subinterval as sample points. Show details carefully.**
**(b) Find the area exactly using the Fundamental Theorem of Calculus (FTC) part 2 and the antiderivative (i.e., integrate directly).**
**Summation Formulas:**
\[
\sum_{i=1}^n i = \frac{n (n+1)}{2}
\]
\[
\sum_{i=1}^n i^2 = \frac{n (n+1)(2n+1)}{6}
\]
\[
\sum_{i=1}^n i^3 = \left[ \frac{n (n+1)}{2} \right]^2
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F99284a24-5b6d-4c49-8bfd-674d8761bb54%2F7e8fbd73-b3f8-41a3-bb58-38b76bd478f9%2Fjtxqb3f_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Calculating the Area under the Curve using Different Methods**
**1. Given the curve \( y = x^2 \) over the interval \([1,3]\):**
**(a) Find the exact value for the area under the curve using the Riemann sums definition of an integral using right endpoints of each subinterval as sample points. Show details carefully.**
**(b) Find the area exactly using the Fundamental Theorem of Calculus (FTC) part 2 and the antiderivative (i.e., integrate directly).**
**Summation Formulas:**
\[
\sum_{i=1}^n i = \frac{n (n+1)}{2}
\]
\[
\sum_{i=1}^n i^2 = \frac{n (n+1)(2n+1)}{6}
\]
\[
\sum_{i=1}^n i^3 = \left[ \frac{n (n+1)}{2} \right]^2
\]
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