find the area under the curve

Question

Find the area under the curve from  to and evaluate it for , . Then find the total area under this curve for .

(a) t = 10  

(b) t = 100  

(c) Total area

Answer 1

Let f(x) = 1.5x - 2.5. To find the area under f(x), we must integrate. Integration of f(x) from x = 7 to x = t yields

∫₇^t f(x) dx = ∫₇^t 1.5x - 2.5 dx

∫₇^t f(x) dx = 0.75x² - 2.5x|₇^t

∫₇^t f(x) dx = 0.75t² - 2.5t - 0.75(7)² + 2.5(7)

∫₇^t f(x) dx = 0.75t² - 2.5t - 36.75 + 17.5

∫₇^t f(x) dx = 0.75t² - 2.5t - 19.25

The solution is as simple as substituting the desired values of t.