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Chapter 7: Problem 6

(a) Let \(f(x):=2\) if \(0 \leq x<1\) and \(f(x):=1\) if \(1 \leq x \leq 2\). Show that \(f \in \mathcal{R}[0,2]\) and evaluate its integral. (b) Let \(h(x):=2\) if \(0 \leq x<1, h(1):=3\) and \(h(x):=1\) if \(1

Short Answer

Expert verified
Both f(x) and h(x) are Riemann-integrable over [0,2] and their integrals are equal to 3.

Step by step solution

01

Verify Riemann Integrability for f(x)

As per the definition, f(x) is well-defined and bounded over the closed interval [0,2]. Hence, f(x) will be Riemann-integrable.
02

Calculate Integral of f(x)

The definite integral of f(x) from 0 to 2 is equal to the sum of the areas of the rectangles under the graph. This can be calculated as:\[\int_0^1 f(x) dx + \int_1^2 f(x) dx = \int_0^1 2 dx + \int_1^2 1 dx = [2x]_0^1 + [x]_1^2 = 2*(1-0) + (2-1) = 2+1 = 3\]
03

Verify Riemann Integrability for h(x)

Similar to f(x), h(x) is also well-defined and bounded over the closed interval [0,2]. Hence, h(x) is also Riemann-integrable.
04

Calculate Integral of h(x)

The definite integral of h(x) from 0 to 2 is similar to part a:\[\int_0^1 h(x) dx + \int_1^2 h(x) dx = \int_0^1 2 dx + \int_1^2 1 dx = [2x]_0^1 + [x]_1^2 = 2*(1-0) + (2-1) = 2+1 = 3\]The only difference is that, at x=1, h(x) is equal to 3, but this doesn't affect the integral since the set of discontinuities has a zero measure in [0,2].

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Most popular questions from this chapter

Suppose that \(f\) is bounded on \([a, b]\) and that there exists two sequences of tagged partitions of \(\\{a, b]\) such that \(\|\dot{\mathcal{P}}\| \rightarrow 0\) and \(\left\|\dot{\mathcal{Q}}_{n}\right\| \rightarrow 0\), but such that \(\lim _{n} S\left(f ; \dot{\mathcal{P}}_{n}\right) \neq \lim _{n} S\left(f ; \mathcal{Q}_{n}\right) .\) Show that \(f\) is not in \(\mathcal{R}[a, b]\).

We have seen in Example \(7.1 .6\) that Thomae's function is in \(\mathcal{R}[0,1]\) with integral equal to \(0 .\) Can the Fundamental Theorem \(7.3 .1\) be used to obtain this conclusion? Explain your answer.

Use the Substitution Theorem \(7.3 .8\) to evaluate the following integrals. (a) \(\int_{0}^{1} t \sqrt{1+t^{2}} d t\), (b) \(\int_{0}^{2} t^{2}\left(1+t^{3}\right)^{-1 / 2} d t=4 / 3\), (c) \(\int_{1}^{4} \frac{\sqrt{1+\sqrt{t}}}{\sqrt{t}} d t\) (d) \(\int_{1}^{4} \frac{\cos \sqrt{t}}{\sqrt{t}} d t=2(\sin 2-\sin 1)\).

If \(f\) and \(g\) are continuous on \([a, b]\) and if \(\int_{a}^{b} f=\int_{a}^{b} g\), prove that there exists \(c \in[a, b]\) such that \(f(c)=g(c)\).
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