Math Problems Vault

Problem 2316

(a)(b)(c)(d)

Determine the sign of each of the following definite integrals:

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(a)

$\displaystyle \int_{0}^{2\pi }x\sin x dx$ ;

(b)

$\displaystyle \int_{0}^{2\pi }\dfrac {\sin x}{x} dx$ ;

(c)

$\displaystyle \int_{-2}^{2}x^{3}e^{x} dx$ ;

(d)

$\displaystyle \int_{\frac{1}{2}}^{1}x^{2}\ln x dx$ .

Problem 2317

(a)(b)(c)

In each of the following, determine which integral is larger:

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(a)

$\displaystyle \int_{0}^{\frac{\pi }{2}}\sin^{10}x dx$ or $\displaystyle \int_{0}^{\frac{\pi }{2}}\sin^{2}x dx$ ?

(b)

$\displaystyle \int_{0}^{1}e^{-x} dx$ or $\displaystyle \int_{0}^{1}e^{-x^{2}} dx$ ?

(c)

$\displaystyle \int_{0}^{\pi }e^{-x^{2}}\cos^{2}x dx$ or $\displaystyle \int_{\pi }^{2\pi }e^{-x^{2}}\cos^{2}x dx$ ?

Problem 2318

(a)(b)(c)(d)

Find the mean value of each given function on the given interval:

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(a)

$f(x)=x^{2}$ on $[0,1]$ ;

(b)

$f(x)=\sqrt{x}$ on $[0,100]$ ;

(c)

$f(x)=10+2\sin x+3\cos x$ on $[0,2\pi ]$ ;

(d)

$f(x)=\sin x\sin (x+\varphi )$ on $[0,2\pi ]$ .

Problem 2319

Find the mean value of the length of the focal radius of the ellipse

r=\dfrac {p}{1-\varepsilon \cos \varphi }\quad (0<\varepsilon <1) .

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Problem 2320

Find the mean value of the velocity of a freely falling body with initial velocity $v_{0}$ .

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Problem 2321

The current intensity varies according to the law

i=i_{0}\sin \left(\dfrac {2\pi t}{T}+\varphi \right) ,

where $i_{0}$ denotes the amplitude, $t$ denotes time, $T$ denotes the period, and $\varphi$ denotes the initial phase. Find the mean value of the square of the current intensity.

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Problem 2321.1

[NO TASK]

Problem 2322

(a)(b)(c)

Let $\displaystyle \int_{0}^{x}f(t) dt=xf(\theta x)$ . Find $\theta$ when:

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(a)

$f(t)=t^{n}$ ( $n>-1$ );

(b)

$f(t)=\ln t$ ;

(c)

$f(t)=e^{t}$ .

What are $\displaystyle \lim_{x\to 0}\theta$ and $\displaystyle \lim_{x\to +\infty }\theta$ ?

Problem 2323

Use the First Mean Value Theorem to estimate the integral: $\displaystyle \int_{0}^{2\pi }\dfrac {dx}{1+0.5\cos x}$ .

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Problem 2324

Use the First Mean Value Theorem to estimate the integral: $\displaystyle \int_{0}^{1}\dfrac {x^{9}}{\sqrt{1+x}} dx$ .

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Problem 2325

Use the First Mean Value Theorem to estimate the integral: $\displaystyle \int_{0}^{100}\dfrac {e^{-x}}{x+100} dx$ .

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Problem 2326

(a)(b)

Prove the equalities

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(a)

(b)

$\displaystyle \lim_{n\to \infty }\int_{0}^{\frac{\pi }{2}}\sin^{n}x dx=0$ .

Problem 2326.1

(a)(b)

$\displaystyle \lim_{n\to \infty }\int_{0}^{1}\dfrac {x^{n}}{1+x} dx=0$ ;

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(a)

(b)

Problem 2327

Suppose the function $f(x)$ is continuous on $[a,b]$ , and $\varphi (x)$ is continuous on $[a,b]$ and differentiable on $(a,b)$ , with

\varphi '(x)\geqslant 0\quad \text{when}\quad a<x<b .

Use integration by parts together with the First Mean Value Theorem to prove the Second Mean Value Theorem.

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Problem 2328

Use the Second Mean Value Theorem to estimate the integral: $\displaystyle \int_{100\pi }^{200\pi }\dfrac {\sin x}{x} dx$ .

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Problem 2329

Use the Second Mean Value Theorem to estimate the integral: $\displaystyle \int_{a}^{b}\dfrac {e^{-ax}}{x}\sin x dx\quad (a\geqslant 0;\ 0<a<b)$ .

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Problem 2330

Use the Second Mean Value Theorem to estimate the integral: $\displaystyle \int_{a}^{b}\sin x^{2} dx\quad (0<a<b)$ .

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Problem 2331

Suppose the functions $\varphi (x)$ and $\psi (x)$ , together with their squares, are integrable on the interval $[a,b]$ . Prove the Cauchy--Bunyakovsky inequality

\left\{ \int _{a}^{b}\varphi (x)\psi (x) dx\right\} ^{2}\leqslant \int _{a}^{b}\varphi ^{2}(x) dx\int _{a}^{b}\psi ^{2}(x) dx .

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Problem 2332

Suppose the function $f(x)$ is continuously differentiable on the closed interval $[a,b]$ and $f(a)=0$ . Prove the inequality

M^{2}\leqslant (b-a)\int _{a}^{b}f'^{2}(x) dx ,

where $\displaystyle M=\sup_{a\leqslant x\leqslant b}\left|f(x)\right|$ .

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Problem 2333

Prove the equality

\lim _{n\to \infty }\int _{n}^{n+p}\dfrac {\sin x}{x} dx=0 .

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