/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 76 Let \(f(x)=c,\) where \(c>0,\... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 76

Let \(f(x)=c,\) where \(c>0,\) be a constant function on \([a, b] .\) Prove that any Riemann sum for any value of \(n\) gives the exact area of the region between the graph of \(f\) and the \(x\) -axis on \([a, b]\).

Short Answer

Expert verified
Question: Prove that the Riemann sum of a constant function \(f(x) = c\), where \(c > 0\), on the interval \([a, b]\) gives the exact area of the region between the graph of \(f\) and the \(x\)-axis. Answer: We proved that the Riemann sum \(R_n = c \cdot (b - a)\) agrees with the integral \(\int_a^b c \, dx = c(b - a)\) for any value of \(n\). This shows that the Riemann sum for a constant function gives the exact area of the region between the graph of the function and the \(x\)-axis on the interval \([a, b]\).

Step by step solution

01

Define the Riemann sum

The Riemann sum for a function \(f(x)\) on the interval \([a, b]\) using \(n\) equal subintervals is given by: \[R_n = \sum_{i = 1}^n f(x_i^*) \cdot \Delta x\] where \(x_i^*\) is a point in the \(i\)-th subinterval and \(\Delta x = (b - a) / n\) is the width of each subinterval.
02

Calculate the Riemann sum for the given constant function

By substituting \(f(x) = c\) into the definition of the Riemann sum, we obtain: \[R_n = \sum_{i = 1}^n c \cdot \Delta x = c \sum_{i = 1}^n \Delta x\]
03

Simplify the Riemann sum

Since \(\Delta x\) is constant, we can factor it out of the sum: \[R_n = c \cdot \Delta x \cdot \sum_{i = 1}^n 1 = c \cdot \Delta x \cdot n\]
04

Express the Riemann sum in terms of \(a\) and \(b\)

Recall that \(\Delta x = (b - a) / n\). We can substitute this back into the Riemann sum: \[R_n = c \cdot \frac{b - a}{n} \cdot n\]
05

Simplify the Riemann sum further

The factors of \(n\) cancel, leaving us with \[R_n = c \cdot (b - a)\]
06

Calculate the integral of the constant function

The integral of \(f(x) = c\) on the interval \([a, b]\) can be calculated as follows: \[\int_a^b c \, dx = c \int_a^b dx = c(x \Big|_a^b) = c(b - a)\]
07

Compare the Riemann sum and the integral

It is clear that the Riemann sum and the integral of the constant function have the same value: \[R_n = c (b - a) = \int_a^b c \, dx\] Therefore, we have proven that the Riemann sum for any value of \(n\) gives the exact area of the region between the graph of the constant function \(f(x) = c\) and the \(x\)-axis on the interval \([a, b]\).

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