Math question

Question

Find the integral of the function using definition of integral. b=6 a=2 (3x²-4x-1)

Thank you

Answer 1

Here is a Desmos graph that actually calculates Riemann sums. Use the slider on n to control number of rectangles. Use the slider on c to use left-hand, midpoint, or right-hand sums.

desmos.com/calculator/89jj4im5ua

Answer 2

Integral of 3x^2 -4x -1 = x^3 -2x^2 - x

evaluated from a=2 to b= 6

= 6^3 - 2(6^2) -6 - (2^3 -2(2^2) -2)

= 216 -72- 6 - (8-8-2)

= 138 +2

= 140

or calculate the area under the curve = the integral

approximate,

divide the area into rectangles, base =1 for 4 rectangles

heights using midpoints = f(5/2), f(7/2), f(9/2), f(11/2)

sum of the rectangles' areas =f(5/2) + f(7/2)+f(9/2)+f(11/2) times base = 1

= 3(5/2)^2 -4(5/2)-1 +3(7/2)^2 -4(7/2) -1 +3(9/2)^2 -4(9/2)-1 + 3(11/2)^2 -4(11/2) -1

= 75/4 - 11 + 147/4 - 15 + 243/4 -19 + 484/4 - 23

= 949/4 - 68

= 237.25 -68

= 149.25 which is fairly close to the actual integral of 140 square units

as the number of rectangles increases, approaching infinity, the rectangles' areas approach 140 as a limit

f(x) = y = 3x^2 -4x -1 is an upward opening parabola

f(2) = 12-8-1 = 3

f(6)= 108-24-1 = 83

draw a line from (2,3) to (6,83)

and calculate the area under the rectangle with triangle on top

= 4x2 + .5(4x80) = 8+160= 168

this is an overestimate as the parabola is curved not a straight line

168 is an upper bound for the integral or area