Question

Find the value of the integral from 2 to 6 of the natural log of x, dx using four rectangles of equal width, with each right end-point used to find the height of each rectangle.

Answer 1

Answer: Choice B.

ln(3) + ln(4) + ln(5) + ln(6)

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Explanation:

Refer to the diagram below. It shows four rectangles each of base 1. Their height varies depending on the output of the natural log function.

More specifically the heights are ln(3), ln(4), ln(5) and ln(6) from left to right. Note how this corresponds exactly to the points on the y = ln(x) curve such that they are the upper right corners of the Riemann Sum rectangles. These corner points are points A,B,C,D.

Since each base is 1 unit wide, this means each rectangle's area is equal to the height. More thoroughly, we say that

area = base*height = 1*f(x) = f(x), where x takes on the values from {3,4,5,6}

Side note: As the diagram shows, we have an overestimate of the actual area under the curve from x = 2 to x = 5.