Alan

The area of the ellipse is the area of the original circle multiplied by the determinant of matrix A.

I'll leave you to do the number crunching.

SUVAT Equations:

Hmm!   -arcsin(x) is the same as arccos(x) to within a constant, so either is acceptable in a solution to the indefinite integral.

As follows:

When theta is very small then sin(theta) → theta, and tan(theta)→theta

So sin(6x)/x + tan(9x) → 6x/x + 9x = 6 + 0 = 6 as x tends to zero

The online calculator at https://www.calculator.net/ has a 1/x button.

Alternatively, use an x^y button (as on the webcalc calculator on the home page) with y as -1.

Total mechanical energy:  E = (1/2)Mv² + (1/2)kx²  where M is mass, v is velocity

At 18cm, assuming we start with zero velocity,  E = (1/2)*1.1*10³*(0.18)^2  Joules

(a) Let xa be the value of x when the PE is 15% of E:   (1/2)k*xa² = 0.15*(1/2)k*(0.18)^2  or xa = sqrt(0.15)*0.18 m

(b) Let va be the velocity at this point.   (1/2)*4*va^2 + (1/2)*1.1*10³*xa² = (1/2)*1.1*10³*(0.18)^2   solve for va (in m/s)

I'll leave you to do the number crunching.

I assume the sections of the top are numbered from 1 to 8.

There are 8*8 = 64 possible outcomes.

There is only one way of getting a total of 16, namely an 8 on the first spin and an 8 on the second spin.  Hence the probability of getting a total of 16 is 1/64.

The ways of getting a total of 5 are:

1st spin   2nd spin

   1            4

   2            3

   3            2

   4            1

That is, there are 4 ways of getting a total of 5.  Hence probability is 4/64 or 1/16.

Thanks all !

"I can't figure out how to solve for ∫ sqrt(2x^2-4) / (7x). Can someone please help me?"

Like so:

(As it is an indefinite integral you should add an arbitrary constant of course.)

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