MAT 265 CALCULUS FOR ENGINEERS I FALL FINAL EXAM 2026 GUIDE (Complete And Verified Study material) (15pages) LEARNEXAMS

Find area of y = f(x) between a ≤ x ≤ b b ● Shortened: ∫ f(x)dx a Examples 3 ● ∫ 2x − 1 2 dx ○ You can either take the area after graphing it out, or take the antiderivative (integral) directly ○ x 2 − 2x ■ The antiderivative ○ (3)2 − 2(3) = 3 ■ Plug in b for x ○ (1)2 − 2(1) = − 1 ■ Plug in a for x ○ (3) − (− 1) = 4 ■ Subtract a from b to find the area between the two points The Integral using limit definition b ● ∫ f(x)dx = lim n ∑ f(xk )Δx a n→∞ k = 1 n ● ∑ 1 = n k=1 ● n n