MAT 265 CALCULUS FOR ENGINEERS I FALL FINAL EXAM 2024 GUIDE Definite Integral 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 ○ x2 − 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(n+1) ∑ k = 2 k=1 n ● ∑ k2 k=1 n(n+1)(2 n+1) 6 =
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(ASU) MAT 265 Calculus For Engineers Fall Final Exam 2024 Guide
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(ASU) MAT 265 Calculus For Engineers Fall Final Exam 2024 Guide(ASU) MAT 265 Calculus For Engineers Fall Final Exam 2024 Guide
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