1. Find ²₀ (|x| - 1)dx , by using analytical, or graphical (hint: review definition of integration) method.

2.Find ²₀ (x + sin(x))dx.

3.A particle moving along the x-axis at the origin x=0, when t=0. A time t, its velocity is v(t) = 1/(2t+1)² which is rapidly decreasing. Show that the particle will never pass the point x=0.5. (Hint: distance travelled at time b, is d(b) = ^b₀ v(t)dt )

4.Since f(x)=1+x² is an increasing function on [1,4], f(1)=2 is the minimum, and f(4)=17 is the maximum of the function f on [1.4].

The error of an integration based on a Riemann sum may be given as the difference between the upper sum and lower sum as,

error(n) = ⁿ_i=1 f(M_i)x_i - ⁿ_i=1 f(m_i)x_i , where f(M_i), and f(m_i) are respectively, the maximum and minimum of the function, f, over the subinterval [x_i-1, x_i],

(a)Show that the error function Error(n) = [f(4) - f(1)]x, when the sub-interval

x_i = x = (4 -1)/n , is of equal length (i.e., divide the interval [1, 4] equally into n subintervals with end points, x₁, x₂, ...., x_i-1,x_i, ...,x_n, where x₁=1, and x_n=4 )

(Hint: The problem becomes easy if you notice the inter relationship between f(M_i) and f(m_i) for i= 1, 2, ...n. Be sure to show your work.)

(b)Compute Error(100) and Error(200). Is error(n) decreasing as n increases from 100 to 200?