Since x is measured from the center, that means that x=0 is the center so the edges of the rod correspond to x=-6 and x=6. that meas that the total charge can be calculated as

$Q=\int^{6}_{-6}2\exp\left(\frac{-|x|}{6}\right)dx$

separating the integral from -6 to 0 and from 0 to 6, taking into account that |x|=-x for x<0 and |x|=x for x >=0, we get $Q=\int^{0}_{-6}2\exp\left(\frac{x}{6}\right)dx+\int^{6}_{0}2\exp\left(\frac{-x}{6}\right)dx$

using the substitution x=-u in the first integral we get $\int^{0}_{-6}2\exp\left(\frac{x}{6}\right)dx=-\int^{0}_{6}2\exp\left(\frac{-u}{6}\right)du=\int^{6}_{0}2\exp\left(\frac{-u}{6}\right)du$

which is the same as the first integral. Thus, the total charge is given by

$Q=2\int^{6}_{0}2\exp\left(\frac{-x}{6}\right)dx$

integrating we get

$Q=4(-6\exp\left(\frac{-x}{6}\right))\big|^{6}_{0}=-24(\exp(-6/6)-\exp(0))=24(1-\exp(-1))$

4 0

3 years ago

3. Consider the following diagram.

yawa3891 [41]

Answer:

m = 3, n = 6

Step-by-step explanation:

The hypotenuse leg (HL) theorem states that two triangles are said to be similar if one leg and the hypotenuse of the triangles are congruent.

From the diagram, comparing the hypotenuse of the triangles;

4m + 1 = 13

4m = 13 - 1

= 12

m = $\frac{12}{4}$

= 3

m = 3

Also, comparing the sides;

2m + n = 8m - 2n

substitute the value of m to have,

6 + n = 24 - 2n

collect like terms,

n + 2n = 24 - 6

3n = 18

n = $\frac{18}{3}$

= 6

n = 6

m = 3, n = 6

6 0

3 years ago