정적분의 뜻과 성질

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 $\frac{\cos ({\pi }^{2})-1}{2\pi }.$

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 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.

[T] Compute the left and right Riemann sums L₁₀ and R₁₀ and their average $\frac{{L}_{10}+{R}_{10}}{2}$ for $f(t)={t}^{2}$ over $[0,1].$ Given that ${\int }_{0}^{1}{t}^{2}dt=0.\overset{-}{33},$ to how many decimal places is $\frac{{L}_{10}+{R}_{10}}{2}$ accurate?

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?

If ${\int }_{1}^{5}\sqrt{1+{t}^{4}}dt=41.7133...,$ what is ${\int }_{1}^{5}\sqrt{1+{u}^{4}}du?$

Estimate ${\int }_{0}^{1}tdt$ using the left and right endpoint sums, each with a single rectangle. How does the average of these left and right endpoint sums compare with the actual value ${\int }_{0}^{1}tdt?$

If f is 1-periodic $(f(t+1)=f(t)),$ odd, and integrable over $[0,1],$ is it always true that ${\int }_{0}^{1}f(t)dt=0?$

If f is 1-periodic and ${\int }_{0}^{1}f(t)dt=A,$ is it necessarily true that ${\int }_{a}^{1+a}f(t)dt=A$ for all A?

Use basic integration formulas to compute the following antiderivatives or definite integrals.

$\int \left(\sqrt{x}-\frac{1}{\sqrt{x}}\right)dx$

Use basic integration formulas to compute the following antiderivatives or definite integrals.

$\int \left({e}^{2x}-\frac{1}{2}{e}^{x/2}\right)dx$