Question

Help calculus module 5 DBQ please show work

Answer 1

1. The four subintervals are [0, 2], [2, 5], [5, 6], and [6, 7]. Their respective right endpoints are 2, 5, 6, and 7. If $C(t)$ denotes the change in sea level $t$ years after 2010, then the total sea level rise over the course of 2010 to 2017 is

$\displaystyle\int_0^7C(t)\,\mathrm dt$

approximated by the Riemann sum,

$C(2)(2-0)+C(5)(5-2)+C(6)(6-5)+C(7)(7-6)\approx\boxed{20\,\mathrm{mm}}$

2. The sum represents the definite integral

$\boxed{\displaystyle\int_1^4\sqrt x\,\mathrm dx}$

That is, we partition the interval [1, 4] into $n$ subintervals, each of width $\dfrac{4-1}n=\dfrac3n$. Then we sample $n$ points in each subinterval, where $1+\dfrac{3k}n$ is the point used in the $k$th subinterval, then take its square root.

3. The integral is trivial:

$\displaystyle\int_1^4\sqrt x\,\mathrm dx=\frac23x^{3/2}\bigg|_{x=1}^{x=4}=\boxed{\frac{14}3}$

4. Using the fundamental properties of the definite integral, we have

$\displaystyle\int_1^4f(x)\,\mathrm dx=e^4-e\implies2\int_1^4f(x)\,\mathrm dx=2e^4-2e$

$\displaystyle\int_1^4(2f(x)-1)\,\mathrm dx=2e^4-2e-\int_1^4\mathrm dx=\boxed{2e^4-2e-3}$

5. First note that $\sec x$ is undefined at $x=\dfrac\pi2$, so the integral is improper. Recall that $(\tan x)'=\sec^2x$. Then

$\displaystyle\int_0^{\pi/2}\sec^2\frac xk\,\mathrm dx=\lim_{t\to\pi/2^-}\int_0^t\sec^2\frac xk\,\mathrm dx$

$=\displaystyle\lim_{t\to\pi/2^-}k\tan\frac xk\bigg|_{x=0}^{x=t}$

$=\displaystyle k\lim_{t\to\pi/2^-}\tan\frac tk$

$=k\tan\dfrac\pi{2k}$

Now,

$k\tan\dfrac\pi{2k}=k\implies\tan\dfrac\pi{2k}=1$

$\implies\dfrac\pi{2k}=\dfrac\pi4+n\pi$

$\implies k=\dfrac2{1+4n}$

where $n$ is any integer.