1.2 The definite integral  (Page 8/16)

$f\left(x\right)=\text{sin}x,a=0,b=2\pi$

$f\left(x\right)=\text{cos}x,a=0,b=2\pi$

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

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

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

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

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

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

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

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

Suppose that $A={\displaystyle {\int }_0^{2\pi }{\text{sin}}^{2}tdt}$ and $B={\displaystyle {\int }_0^{2\pi }{\text{cos}}^{2}tdt}.$ Show that $A+B=2\pi$ and $A=B\text{.}$

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

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

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

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

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

Suppose $\left[a,b\right]$ can be subdivided into subintervals $a=a_0<a_1<a_2<\text{⋯}<a_N=b$ such that either $f\ge 0$ over $\left[a_{i-1},a_i\right]$ or $f\le 0$ over $\left[a_{i-1},a_i\right].$ Set $A_i={\displaystyle {\int }_{a_{i-1}}^{a_i}f\left(t\right)dt}.$

1. Explain why ${\displaystyle {\int }_a^{b}f\left(t\right)dt}=A_1+A_2+\text{⋯}+A_N.$
2. Then, explain why $\left|{\displaystyle {\int }_a^{b}f\left(t\right)dt}\right|\le {\displaystyle {\int }_a^b\left|f\left(t\right)\right|dt}.$

Suppose f and g are continuous functions such that ${\displaystyle {\int }_c^{d}f\left(t\right)dt}\le {\displaystyle {\int }_c^{d}g\left(t\right)dt}$ for every subinterval $\left[c,d\right]$ of $\left[a,b\right].$ Explain why $f\left(x\right)\le g\left(x\right)$ for all values of x .

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

Suppose that $\left[a,b\right]$ can be partitioned. taking $a=a_0<a_1<\text{⋯}<a_N=b$ such that the average value of f over each subinterval $\left[a_{i-1},a_i\right]=1$ is equal to 1 for each $i=1\text{,…,}N.$ Explain why the average value of f over $\left[a,b\right]$ is also equal to 1.

Suppose that for each i such that $1\le i\le N$ one has ${\displaystyle {\int }_{i-1}^{i}f\left(t\right)dt}=i.$ Show that ${\displaystyle {\int }_0^{N}f\left(t\right)dt}=\frac{N\left(N+1\right)}{2}.$

Suppose that for each i such that $1\le i\le N$ one has ${\displaystyle {\int }_{i-1}^{i}f\left(t\right)dt}=i^2.$ Show that ${\displaystyle {\int }_0^{N}f\left(t\right)dt}=\frac{N\left(N+1\right)\left(2N+1\right)}{6}.$

[T] Compute the left and right Riemann sums L ₁₀ and R ₁₀ and their average $\frac{L_{10}+R_{10}}{2}$ for $f\left(t\right)=t^2$ over $\left[0,1\right].$ Given that ${\displaystyle {\int }_0^1{t}^{2}dt}=0.\overline{33},$ to how many decimal places is $\frac{L_{10}+R_{10}}{2}$ accurate?

[T] Compute the left and right Riemann sums, L ₁₀ and R ₁₀ , and their average $\frac{L_{10}+R_{10}}{2}$ for $f\left(t\right)=\left(4-t^2\right)$ over $\left[1,2\right].$ Given that ${\displaystyle {\int }_1^2\left(4-t^2\right)dt}=1.\overline{66},$ to how many decimal places is $\frac{L_{10}+R_{10}}{2}$ accurate?

If ${\displaystyle {\int }_1^5\sqrt{1+t^4}dt}=\mathrm{41.7133..}.,$ what is ${\displaystyle {\int }_1^5\sqrt{1+u^4}du}?$

Estimate ${\displaystyle {\int }_0^{1}tdt}$ 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 ${\displaystyle {\int }_0^{1}tdt}?$

Estimate ${\displaystyle {\int }_0^{1}tdt}$ by comparison with the area of a single rectangle with height equal to the value of t at the midpoint $t=\frac{1}{2}.$ How does this midpoint estimate compare with the actual value ${\displaystyle {\int }_0^{1}tdt}?$

From the graph of $\text{sin}\left(2\pi x\right)$ shown:

1. Explain why ${\displaystyle {\int }_0^1\text{sin}\left(2\pi t\right)dt}=0.$
2. Explain why, in general, ${\displaystyle {\int }_a^{a+1}\text{sin}\left(2\pi t\right)dt}=0$ for any value of a .
   

If f is 1-periodic $(f\left(t+1\right)=f\left(t\right)),$ odd, and integrable over $\left[0,1\right],$ is it always true that ${\displaystyle {\int }_0^{1}f\left(t\right)dt}=0?$

If f is 1-periodic and ${\displaystyle {\int }_0^{1}f\left(t\right)dt}=A,$ is it necessarily true that ${\displaystyle {\int }_a^{1+a}f\left(t\right)dt}=A$ for all A ?