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Integral Calculus and Position of an Object

1. R is the region bounded by the curve y = x^2 -2x and the x axis. Sketch the given region R and then find its area.

2. Using the following integral and the facts that v(t) = x'(t) at time t of an object moving along the x axis is given, along with the initial position x(0) of the object. Answer the questions a - c.

x'(t) = 3(5-2t)^3 ; x(0)=5

a) The position x(t) at time t?
b) The position of the object at time t = 4?
c) The time when the object is at x = 3? Hint: (-3/8(5-2t)^4 +239.3 =3 )

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The intersection points are (0 , 0) and (2 , ...

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Solution to the posted problem is given with step by step explanatioin in such a way that student could easily understand the concept and procedure to solve such type of problems. To make the concept clear a graph is also included in the solution. The complete detailed solution is given in the attached solution file.