Question

Consider the function $f(x) = 6x$ over the interval $[0, 7]$. Divide the interval into $n$ subintervals of equal length. How long is each subinterval? Length is In order to determine an overestimate for the area under the graph of the function, at what x-value should you evaluate $f(x)$ to determine the height of the first rectangle? x = Find a formula for the x-value in the kth subinterval which determines the height of the kth rectangle. x = Write down a Riemann sum for $f(x)$ over the given interval using the x-values you calculated above. Riemann sum is $sum_{k=1}^{n}$ Using the formula $sum_{k=1}^{n} k = frac{n(n+1)}{2}$, write down the above Riemann sum without using a $sum$. Riemann sum is Compute the limit of the above sum as $n o infty$. The limit is

          Consider the function $f(x) = 6x$ over the interval $[0, 7]$.
Divide the interval into $n$ subintervals of equal length. How long is each subinterval?
Length is 
In order to determine an overestimate for the area under the graph of the function, at what x-value should you evaluate $f(x)$ to determine the height of
the first rectangle?
x = 
Find a formula for the x-value in the kth subinterval which determines the height of the kth rectangle.
x = 
Write down a Riemann sum for $f(x)$ over the given interval using the x-values you calculated above.
Riemann sum is $sum_{k=1}^{n}$
Using the formula $sum_{k=1}^{n} k = frac{n(n+1)}{2}$, write down the above Riemann sum without using a $sum$.
Riemann sum is 
Compute the limit of the above sum as $n 	o infty$.
The limit is

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$Consider the function f(x) = 6x over the interval [0, 7]. Divide the interval into n subintervals of equal length. How long is each subinterval? Length is In order to determine an overestimate for the area under the graph of the function, at what x-value should you evaluate f(x) to determine the height of the first rectangle? x = Find a formula for the x-value in the kth subinterval which determines the height of the kth rectangle. x = Write down a Riemann sum for f(x) over the given interval using the x-values you calculated above. Riemann sum is sumk=1^n Using the formula sumk=1^n k = fracn(n+1)2, write down the above Riemann sum without using a sum. Riemann sum is Compute the limit of the above sum as n o infty. The limit is$

Added by Matthew S.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sri K

08/07/2022

Step 1

Step 1: The length of each subinterval is given by \( \frac{7}{n} \) where \( n \) is the number of subintervals. Show more…

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Consider the function f(x) = 6x over the interval [0, 7]. Divide the interval into n subintervals of equal length. How long is each subinterval? Length is In order to determine an overestimate for the area under the graph of the function, at what x-value should you evaluate f(x) to determine the height of the first rectangle? x = Find a formula for the x-value in the kth subinterval which determines the height of the kth rectangle. x = Write down a Riemann sum for f(x) over the given interval using the x-values you calculated above. Riemann sum is Using the formula sum k = n(n+1)/2, write down the above Riemann sum without using a sum. Riemann sum is Compute the limit of the above sum as n -> infinity. The limit is

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00:01 What this problem is really asking us for is to do the steps for the remand sum.

00:07 Now, for the first part, they are asking us what is the length of the intervals, assuming we divide into ends of intervals.

00:15 They're not telling us how many intervals this n is.

00:18 So, well, let's do first the graph.

00:20 It will help us understand.

00:22 We go like this, and the graph, well, the graph for the function f of x equals 6x is a line.

00:30 Like this with about that slope and what we are really doing is so if this is ceto and this is seven we are dividing this into certain amount of rectangles which we don't really know what are they but like they're not telling us what are these of intervals but this is kind of like what we're doing like this like this and let's do one more and basically calculating the area of each of this rectangle of course the smaller these rectangles there are the more accurate the area is.

01:04 So this ends of intervals is just finding the difference between the x values.

01:08 So 7 minus 0 and divided by the amount of sub intervals we have.

01:14 But since the problem is not really specifying, we just write n because that's what the problem is saying n's of intervals.

01:20 So the length of each of these subintervals is 7 over n.

01:26 This stands for part a.

01:27 For part b, they want us to find the height of these rectangles, if we want, of the first rectangle, if we want to overestimate the calculation.

01:37 To overestimate, there are three options we have.

01:39 For each of these rectangles, we can find the area at the first, at the middle, or at the last point of the rectangle.

01:48 For overestimating, we choose the leftmost point, the rightmost point.

01:55 So this one, the last point of the rectangle.

01:58 But they want us to find what is the x value for the first rectangle...

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