Find the area under the curve

Question

Use five rectangles to approximate the area under the curve f(x)=0.4x^3-2.1x^2+20 from x=1 to x=7.

The answer is 102.89. I don't understand how my textbook gets this answer, I either get 120.6 or 178.8 sq units can anyone explain this problem to me?

Answer 1

You can solve this using a left Riemann sum, where your sections are of length 1.2. You get the section length by dividing your total interval (7-1 = 6) by the rectangle count you need (5) to get 1.2.

Then, we find the individual sections by repeatedly adding 1.2 to 1 (the left interval) until we reach the right interval (we ignore the 7 since it's a left Riemann sum): 1, 2.2, 3.4, 4.6, and 5.8.

Then we plug each of these numbers into the function to get the rectangles' height for the approximation: 18.3, 14.1, 11.4, 14.5, and 27.4.

We then multiply each of the heights of the rectangles by the width (1.2, the section length from before) and sum them up to get the total approximate area: (1.2 • 18.3) + (1.2 • 14.1) + (1.2 • 11.4) + (1.2 •14.5) + (1.2 • 27.4) = 102.84. The slight deviance from the textbook's answer probably comes from rounding.