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
`\frac{4-1}n=\frac3n`. Then we sample
`n` points in each subinterval, where
`1+\frac{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=\frac\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\frac\pi{2k}`

Now,

`k\tan\frac\pi{2k}=k\implies\tan\frac\pi{2k}=1`

`\implies\frac\pi{2k}=\frac\pi4+n\pi`

`\implies k=\frac2{1+4n}`

where
`n` is any integer.