1. ## Integration, Maclaurin series, and radioactivity

integrate:-

1. integral (4x-x^2) ^ (1/2)from 0 to 4
i know that I should use completing the square
so i ended up with (-4-(x-2)^2)^(1/2)

however, it doesn't look right to me, May I get a helpful solution?

2.
A certain radioactive isotope is observed to decay to 98% of its initial amount over a
period of one year.
a) Assume that the sample has an initial mass of 100g. Find a function that represents
the mass of a sample as a function of time (in years).
b) What is the half-life of the isotope?
c) How long will it take the sample with initial mass of 100g to decay to a mass of 8g?

3.
Find the Maclaurin series for the functions sinh(
x) and cosh(x) by using the Maclaurin
series for
ex and the de nitions of sinh(x) and cosh(x) in terms of ex. Compute the radius

of convergence for each series.

2. Originally Posted by Sally_Math
integrate:-

1. $integral (4x-x^2) ^ (1/2)$ from 0 to 4
i know that I should use completing the square
so i ended up with $(-4-(x-2)^2)^(1/2)$

however, it doesn't look right to me, May I get a helpful solution?

This makes no sense to me, sorry.

Originally Posted by Sally_Math
2.
A certain radioactive isotope is observed to decay to 98% of its initial amount over a
period of one year.
a) Assume that the sample has an initial mass of 100g. Find a function that represents
the mass of a sample as a function of time (in years).
b) What is the half-life of the isotope?
c) How long will it take the sample with initial mass of 100g to decay to a mass of 8g?
Try $M = 100\times 0.98^t$ where $t$ is years and $M$ is mass.

For half life sub $M=50$

3. integral (4x-x^2) ^ (1/2)from 0 to 4
i know that I should use completing the square
so i ended up with (-4-(x-2)^2))^(1/2)

4. Originally Posted by Sally_Math
integral (4x-x^2) ^ (1/2)from 0 to 4
i know that I should use completing the square
so i ended up with (-4-(x-2)^2))^(1/2)
No, that "4" cannot possibly be negative.

$4x- x^2= -(x^2- 4x)$. To 'complete the square' you must add and subtract $(4/2)^2= 4$: [tex]4x- x^2= -(x^2- 4x+ 4- 4)= -(-4)- (x- 2)^2= 4- (x- 2)^2.

That is $\int \sqrt{4x- x^2}dx= \int \sqrt{4- (x-2)^2}dx$.

Try a trig substitution.

5. Originally Posted by Sally_Math
integrate:-

1. integral (4x-x^2) ^ (1/2)from 0 to 4
i know that I should use completing the square
so i ended up with (-4-(x-2)^2)^(1/2)

however, it doesn't look right to me, May I get a helpful solution?

2.
A certain radioactive isotope is observed to decay to 98% of its initial amount over a
period of one year.
a) Assume that the sample has an initial mass of 100g. Find a function that represents
the mass of a sample as a function of time (in years).
b) What is the half-life of the isotope?
c) How long will it take the sample with initial mass of 100g to decay to a mass of 8g?

(a) Let Q(t) be the quantity of isotope, in g, after t years. Saying that it decays at a steady rate means that $\frac{dQ}{dt}= kQ$ for some (negative) number k. Rewrite that as $\frac{dQ}{Q}= kdt$ and integrate both sides. Solving for Q will give an equation involving ln(Q) which gives Q as an exponential function of t involving the unknown value k and the integration constant C.

Use the fact that Q(0)= 100 and Q(1)= .98(100)= 98 to find k and c.

(b)The "half life" is the time it take to decay to half the original amount. Since the original amount was 100 g, half is 50 g. Set the function in (a) equal to 50 and solve for t.

(c) Set the function you found in (a) equal to 8 and solve for t.

3.
Find the Maclaurin series for the functions sinh(
x) and cosh(x) by using the Maclaurin
series for
ex and the de nitions of sinh(x) and cosh(x) in terms of ex. Compute the radius

of convergence for each series.
What are the definitions of sinh(x) and cosh(x) in terms of $e^x$?