Set up the integral for the surface area obtained by rotating the curve about the given axis. (i)  y=ln(1+x)  from (0,0) to (2, ln(3) ) about the x-axis

Set up the integral for the surface area obtained by rotating the curve about the given axis.   (i) y=e^x-x from (0,1) to (3,e^3-3) about the y-axis.

  (i)    Find the moment Mx = [answer1] (ii)   Find the moment My = [answer2] (iii)  Find the total mass m = [answer3] (iv)  Find the center of mass  = ( [answer4] ). Round your answers to two decimal places. Answer (iv) by using a comma then a single space to separate answers.

  (i)    Find the moment Mx = [answer1] (ii)   Find the moment My = [answer2] (iii)  Find the total mass m = [answer3] (iv)  Find the center of mass  = ( [answer4] ). Round your answers to two decimal places. Answer (iv) by using a comma then a single space to separate answers.

A triangular tank measuring 12m tall, 15m long, and 4m across contains water 6m deep. Which of the following integrals represents the work required to pump all the water out of the top of the tank? (Assume the acceleration due to gravity is 10 m/s\(^2\) and the density of water is 100 kg/m\(^3\).)  

Uranium 238 is a radioactive material with applications in nuclear technology. It decays exponentially with a half-life of about 4.5 billion years. If given an initial mass of 5 grams, find the mass that remains after 3 billion years. Note: \(\exp(x)\) means \(e^x\) so for example, \(\exp(5) = e^5.\)  

The table below shows values of a function \(f(x)\).  Use Trapezoid Rule with four equal subintervals (\(n=4\)) to estimate the value of the integral \[\displaystyle \int_1^5 f(x) \; dx.\] 

Evaluate the integral, if it exists.\[\int_1^\infty \frac{4}{x^3} \;dx\]

Suppose the cost of utilities for a house is given by \[C(t) = 50 \cos\left(\frac{\pi}{6}t\right) + 100,\] where \(t\) is in months. Find the average cost of utilities from December (\(t=0\)) to June (\(t=6\)).

Find the equation of the tangent line to the curve \[y = 3 \arctan(2x)\] at the point \(\displaystyle x = \frac{1}{2}\).