Find the area under the curve of y = −4x + 20 between x = −2 and x = 4 using
geometry and then again using the fundamental theorem of calculus.

Question

Find the area under the curve of y = −4x + 20 between x = −2 and x = 4 using
geometry and then again using the fundamental theorem of calculus.

anonymous · Answer

I was going to draw this but it wont work. Anyway, if you draw the graph of the line from x=-2 to x=4, you can use geometry to find the area. The top triangle is 6 wide by 24 tall giving you 72 for area (1/2*6*24). The bottom box is 6 by 4, area=24. Then the total area is 96. Using calculus you have to integrate, from x=-2 to x=4:$$area = \int\limits_{-2}^{4} (-4x+20)dx$$

Solving the integral you get
$$area = 4(-x^{2}/2 + 5x) $$ evaluated from x=-2 to x=4.
Which gives you 4(-8+20) - 4(-2-10) = 48+48 = 96