Use the Fundamental Theorem of Calculus to find the "area under curve" of f ( x ) = 6 x + 4 between x = 1 and x = 6 .

Question

Use the Fundamental Theorem of Calculus to find the "area under curve" of  between  and .

is my lower limit 1 and upper limit 6? I tried graphing the equation on desmos but it's hard to calculate the area of each rectangle because the slope is so steep. also, would I use a left or right approximation?

Answer 1

Between x = 1 and x = 6, the graph of f(x) = 6x + 4 lies above the x-axis.

So, the area below the graph of y = f(x) and above the x-axis between x = 1 and x = 6 is given by:

Area = ∫_{(from 1 to 6)} (6x + 4) dx = (3x² + 4x)_{(from 1 to 6)} = 132 - 7 = 125

Answer 2

Bri

It's not clear how to use the FTC to find this as both limits are given as constants.

It's straightforward to calculate it geometrically exactly because if you draw the area under the line between x=1 and x=6 and the x axis, you get a trapezium with a = 6x1+4 = 10, b = 6x6 + 4 = 40, and h = 6-1 = 5.

So you can use a formula for area of trapezium to find the integral. No approximations needed.

Mike