적분 전체 문제

In the following exercises, find the average value f_ave of f between a and b, and find a point c, where $f(c)={f}_{\text{ave}}.$

$f(x)=(3-|x|),a=-3,b=3$

In the following exercises, find the average value f_ave of f between a and b, and find a point c, where $f(c)={f}_{\text{ave}}.$

$f(x)=\sin x,a=0,b=2\pi$

In the following exercises, find the average value f_ave of f between a and b, and find a point c, where $f(c)={f}_{\text{ave}}.$

$f(x)=\cos x,a=0,b=2\pi$

In the following exercises, approximate the average value using Riemann sums L₁₀₀ and R₁₀₀. How does your answer compare with the exact given answer?

[T] $y=\ln (x)$ over the interval $[1,4];$ the exact solution is $\frac{\ln (256)}{3}-3.$

In the following exercises, approximate the average value using Riemann sums L₁₀₀ and R₁₀₀. How does your answer compare with the exact given answer?

[T] $y={e}^{x/2}$ over the interval $[0,1];$ the exact solution is $2\left(\sqrt{e}-1\right).$

In the following exercises, approximate the average value using Riemann sums L₁₀₀ and R₁₀₀. How does your answer compare with the exact given answer?

[T] $y=\tan x$ over the interval $\left[0,\frac{\pi }{4}\right];$ the exact solution is $\frac{2\ln (2)}{\pi }.$

In the following exercises, approximate the average value using Riemann sums L₁₀₀ and R₁₀₀. How does your answer compare with the exact given answer?

[T] $y=\frac{x+1}{\sqrt{4-{x}^{2}}}$ over the interval $[-1,1];$ the exact solution is $\frac{\pi }{6}.$

In the following exercises, compute the average value using the left Riemann sums L_N for $N=1,10,100.$ How does the accuracy compare with the given exact value?

[T] $y={x}^{2}-4$ over the interval $[0,2];$ the exact solution is $-\frac{8}{3}.$

In the following exercises, compute the average value using the left Riemann sums L_N for $N=1,10,100.$ How does the accuracy compare with the given exact value?

[T] $y=x{e}^{{x}^{2}}$ over the interval $[0,2];$ the exact solution is $\frac{1}{4}({e}^{4}-1).$

In the following exercises, compute the average value using the left Riemann sums L_N for $N=1,10,100.$ How does the accuracy compare with the given exact value?

[T] $y={\left(\frac{1}{2}\right)}^{x}$ over the interval $[0,4];$ the exact solution is $\frac{15}{64\ln (2)}.$

In the following exercises, compute the average value using the left Riemann sums L_N for $N=1,10,100.$ How does the accuracy compare with the given exact value?

[T] $y=x\sin ({x}^{2})$ over the interval $[-\pi ,0];$ the exact solution is $\frac{\cos ({\pi }^{2})-1}{2\pi }.$

In the following exercises, compute the average value using the left Riemann sums L_N for $N=1,10,100.$ How does the accuracy compare with the given exact value?

Suppose that $A={\int }_{0}^{2\pi }{\sin }^{2}tdt$ and $B={\int }_{0}^{2\pi }{\cos }^{2}tdt.$ Show that $A+B=2\pi$ and $A=B.$

In the following exercises, compute the average value using the left Riemann sums L_N for $N=1,10,100.$ How does the accuracy compare with the given exact value?

Suppose that $A={\int }_{-\pi /4}^{\pi /4}{\sec }^{2}tdt=\pi$ and $B={\int }_{-\pi /4}^{\pi /4}{\tan }^{2}tdt.$ Show that $A-B=\frac{\pi }{2}.$

Show that the average value of ${\sin }^{2}t$ over $[0,2\pi ]$ is equal to 1/2 Without further calculation, determine whether the average value of ${\sin }^{2}t$ over $[0,\pi ]$ is also equal to 1/2.

Show that the average value of ${\cos }^{2}t$ over $[0,2\pi ]$ is equal to $1/2.$ Without further calculation, determine whether the average value of ${\cos }^{2}(t)$ over $[0,\pi ]$ is also equal to $1/2.$

Explain why the graphs of a quadratic function (parabola) $p(x)$ and a linear function $\ell (x)$ can intersect in at most two points. Suppose that $p(a)=\ell (a)$ and $p(b)=\ell (b),$ and that ${\int }_{a}^{b}p(t)dt>{\int }_{a}^{b}\ell (t)dt.$ Explain why ${\int }_{c}^{d}p(t)>{\int }_{c}^{d}\ell (t)dt$ whenever $a\le c<d\le b.$

Suppose that parabola $p(x)=a{x}^{2}+bx+c$ opens downward $(a<0)$ and has a vertex of $y=\frac{-b}{2a}>0.$ For which interval $[A,B]$ is ${\int }_{A}^{B}(a{x}^{2}+bx+c)dx$ as large as possible?

Suppose $[a,b]$ can be subdivided into subintervals $a={a}_{0}<{a}_{1}<{a}_{2}<\cdots <{a}_{N}=b$ such that either $f\ge 0$ over $[{a}_{i-1},{a}_{i}]$ or $f\le 0$ over $[{a}_{i-1},{a}_{i}].$ Set ${A}_{i}={\int }_{{a}_{i-1}}^{{a}_{i}}f(t)dt.$

a. Explain why ${\int }_{a}^{b}f(t)dt={A}_{1}+{A}_{2}+\cdots +{A}_{N}.$ b. Then, explain why $|{\int }_{a}^{b}f(t)dt|\le {\int }_{a}^{b}|f(t)|dt.$

Suppose the average value of f over $[a,b]$ is 1 and the average value of f over $[b,c]$ is 1 where $a<c<b.$ Show that the average value of f over $[a,c]$ is also 1.

Suppose that $[a,b]$ can be partitioned. taking $a={a}_{0}<{a}_{1}<\cdots <{a}_{N}=b$ such that the average value of f over each subinterval $[{a}_{i-1},{a}_{i}]=1$ is equal to 1 for each $i=1,\ldots ,\,N.$ Explain why the average value of f over $[a,b]$ is also equal to 1.