Question

I'm struggling on calculus. PLEASE HELP ASAP. use red writing to help out.​

Answer 1

Step-by-step explanation:

1. Given: a = 4t + 4

We know that

$a = \frac{dv}{dt} \: \: or \: dv = adt$

By definition, the integral of a power function x^n is

$\int {x}^{n} dx = \frac{ {x}^{n + 1} }{n + 1} + k$

Integrating the acceleration a, we get

$v = \int adt = \int(4t + 4)dt = 2 {t}^{2} + 4t + k$

where k = constant of integration. We know that v = 10 when t = 0 so when we do the substitution, we get k = 10 therefore, the final expression for v is

$v = 2 {t}^{2} + 4t + 10$

To find s, we need to integrate v. Knowing that

$v = \frac{ds}{dt} \: \: or \: s = \int v \: dt$

$s = \int(2 {t}^{2} + 4t + 10)dt$

$= \frac{2}{3} {t}^{3} + 2 {t}^{2} + 10t + k$

where k once again is the constant of integration. We know that s = 2 when t = 0, which gives us k = 2. Therefore, the final expression for s is

$s = \frac{2}{3} {t}^{3} + 2 {t}^{2} + 10t + 2$

2. The potential difference V between two boundaries a and b is given by

$V = \frac{q}{2\pi \epsilon_0 \epsilon_r} \int_{b}^{a} \frac{dr}{r}$

Note that the integral in the expression above can be rewritten and the integrated as

$\int_{b}^{a} \frac{dr}{r} =\int_{b}^{a} {r}^{ - 1} dr = \ln |a| - \ln |b|$

so the potential difference V is then J

$V = \frac{q}{2\pi \epsilon_0 \epsilon_r} \int_{b}^{a} \frac{dr}{r}$

$= \frac{2 \times {10}^{ - 6} }{2\pi (8.85\times {10}^{ - 12}) (2.77)}( \ln |20| - \ln |10| )$

$= 9000 \: V$