Calculus Answers

Obtain the Fourier series for the following function

f(x)=0 when -π<x<0

f(x)=sinx when 0<x<π

Find the relative extreme values of the function f (x) = x3 − x2 − 6x + 5.

any function with domain PXP is a binary function

Evaluate integral of (z-3)^4 where c is the circle |z-3|=4

Find the surface area of the function x= ^3 square root of y between x=0 and x=1 when revolved around the axis

Find the volume generated if the function y= square root of 1-x^2 is revolved around the x-axis (use only the positive area)

Use the driven equation x (t) from (C) and solve it using the following

numerical methods over the time interval 0 ≤ t ≤ 3 seconds at h=0.5.

I. Using the trapezium method

II. Using a Simpsons rule

2- Using a mathematical model and calculus methods (e.g. numerical and

integration methods) to solve given engineering problem (Eq. 1).

Your tasks is

d) Using definite integration and driven equation (from c) to find the area

under curve over the time interval 0 ≤ t ≤ 3 seconds.

e) Using a mid-ordinate rule and driven equation (from c) to find the area

under curve over the time interval 0 ≤ t ≤ 3 seconds at h= 0.5.

f) Find an accurate mathematical model (e.g. equation) to correlate

position and time. To complete this task you should be able to sketch

the graph again, find the accurate equation using an excel sheet and

trendline.

g) Compare the R2 between the new and previous equations (step c and f).

h) Use the driven equation x (t) from (C) and solve it using the following

numerical methods over the time interval 0 ≤ t ≤ 3 seconds at h=0.5.

I. Using the trapezium method

II. Using a Simpsons rule

Problem 1: A force of 500 dynes stretches a spring from its natural length of 20 cm to a length of 24 cm. Find the work done in stretching the spring from its natural length to a length of 28 cm.

Problem 2: A spring has a natural length of 6 in. A 1200-lb force compresses it to 5 1/2 in. Find the work done in compressing it from 6 in to 4 1/2 in.

Problem 3: An upright right-circular cylindrical tank of radius 5 ft and height 10 ft is filled with water.

(a) How much work is done by pumping the water to the top of the tank?

(b) Find the work required to pump the water to a level of 4 ft above the top of the tank.

Problem 4: A conical reservoir 10m deep and 8m across the top is filled with water to a depth of 5m.

The reservoir is emptied by pumping the water over the top edge. How much work is done

in the process?

Centroid of Plane Area

Locate the centroid of the region 𝑅 bounded by the given curves. Sketch the graph.

1.) 𝑅:𝑦=𝑥^2, 𝑦=2𝑥^2 −3𝑥

2.) 𝑅:𝑦^2=4𝑥, 𝑦=4, 𝑦−𝑎𝑥𝑖𝑠

3.) 𝑅:𝑥=2𝑦−𝑦^2, 𝑦−𝑎𝑥𝑖𝑠

4.) 𝑅:𝑦=2𝑥+1, 𝑥+𝑦=7, 𝑥=8