For the following: a) Calculate the values of these integrals (A and B) by hand.

Question

For the following:

a) Calculate the values of these integrals (A and B) by hand. (Please explain your work :))

b) For A, the left sum should be less than the exact solution while the right side sum should be greater than the exact solution. Explain this by plotting and sketching your graph by hand.

c) Repeat part b for B. Is the same true? (left=less than exact solution, right=more than exact solution)

d) For A and B, is 1000 "slices/sections" for the right and left endpoints approximately within 10% of the exact answer?

Note: For part A, please note that the "most accurate" general antiderivative of lnx is xlnx-x+C

Explanation / Answer

Solution: (a)

(A) (1 to 2) ln x dx

Apply integration by parts;

uv' = uv - u'v

Let u = ln(x), u' = 1/x, v' = 1, v = x

So   ln(x) dx = x ln(x) - (1/x) * x dx

= {x ln(x) - x} limit from (1 to 2)

= {2 ln(2) - 2} - {ln(1) - 1}

= ln(4) - 1

(B) (1 to 2) sin (50x) dx

compute indefinate integral;

Apply integral substitution;

Let u = 50x, du = 50 dx => dx = (1/50) du

= (1/50) sin(u) du

= (1/50) (-cos(u))

substitute back u = 50x

= - (1/50) cos(50x)

Compute boundries;

(1 to 2) sin (50x) dx = - (1/50) cos(50x) (from 1 to 2)

= -(1/50) {cos(100) - cos(50)}

= (1/50) { cos(50) - cos(100) }

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