# Recent questions and answers in Calculus

Suppose Aaron is pumping water into a tank (in the shape of an inverted right circular cone) at a rate of 1600 ft^3/min.
If the altitude is 10 ft and the radius of the base is 5 ft, find the rate at which the radius is changing when the height of the water is 7 ft.
Find y(1) if

$\frac{dy}{dx} = x y^2 \text { and } y(0) =1.$

Solve equation

$e^x - e^{-x} = e$

The area of an equilateral triangle is increasing at the rate of 5 m^2/hr.

Find the rate at which the height is changing when the area is

$\frac{64}{\sqrt3} m^2$

Suppose that f is a function that has a continuous second derivative and that satisfies

$f(0) = 4, f(1) =3, f'(0) = 5, f'(1) = 7, f''(0) =8 \text{ and } f''(1) = 11.$

Show that

$\int_0^1f(x) f''(x)dx \leq1$

Show that for any positive continuous function f on [0, a]

$\int_{0}^{a} \frac{f(x)}{f(x) + f(a-x)} dx = a/2$

Find the integral

$\int x^3 \sqrt{9-x^2} dx$

Solve the following IVP.

$y^{3}-5y''-22y'+56y=0$

y(0)=1

y'(0)=-2

y''(0)=-4

Find the arc length of the function over the indicated interval.

$x=\frac {1}{3}\sqrt {y}(y-3)$

$1\leq y\leq 4$

Find the arc length of the function over the indicated interval.

$x=\frac {1}{3}(y^2+2)^{3/2}$

$0\leq y\leq 4$

Find the arc length of the function over the indicated interval.

$y=ln\left ( \frac{e^x+1}{e^x-1} \right )$

[ln 2,ln 3]

Find the arc length of the function over the indicated interval.

$y=\frac {1}{2}(e^x+e^{-x})$

[0,2]

Find the arc length of the function over the indicated interval.

$y=ln (sinx)$

$\left [ \pi/4,3\pi/4 \right ]$

Find the arc length of the function over the indicated interval.

$y=ln (sinx)$

$\left [ \pi/4,3\pi/4 \right ]$

Find the arc length of the function over the indicated interval.

$y=\frac {x^5}{10}+\frac {1}{6x^3}$

[2,5]

Find the arc length of the function over the indicated interval.

$y=\frac {x^4}{8}+\frac {1}{4x^2}$

[1,3]

$y=\left\{\begin{matrix} \frac {sinx}{x}\; \; \; x>0\\ 1 \; \; \; x=0 \end{matrix}\right.$
$y=0$
$x=0$
$x=\pi$